A discussion of this subject should map the variants spoken in São Paulo state versus the density of Italian immigration. Do the areas where Italian immigration was heavier speak differently from the areas were it was lighter?
I haven't seen a study that does this. So, with no systematical data at hand, I would contrast the Portuguese spoken in São Paulo city to that spoken in Campinas (the state's second city by population). Both cities received huge numbers of Italian immigrants. But Campinas' Portuguese is similar to that of other cities in the hinterland (for instance, in the southwestern cities, where Italian immigration was smaller, and indirect, and where Japanese immigration was also important). São Paulo's Portuguese, in contrast, is remarkably different from that of the hinterland. And so I would treat the claim of Italian influences in the Portuguese spoken in São Paulo with some caution. Below, I will underline the reasons I take the influences outlined by Marcílio Melo Vieira's thesis with some salt.
The pronunciation of postvocalic "r"
Brazilian Portuguese can be divided roughly into two variants, concerning the pronunciation of postvocalic "r":
A. In the litoral, from somewhere in between Santos and Rio de Janeiro to the North, the "r" in "carta" is pronounced similar to "rr" as in "carro".
B. To the South, and then again to the North through the hinterland, as far North as the Amazon, it is pronounced similar to "r" as in "caro".
Variant A is spoken in Rio de Janeiro; in São Paulo - all of the state, not just the city - B is widely predominant.
But then those variants can be subdivided, regarding what exactly is an "r" or an "rr". In Rio de Janeiro, "carta" is pronounced [ˈkaʁta], as "carro" is pronounced ['kaʁu]]; but in other places we can hear [ˈkaxta] or [ˈkahta], and I fear a few other pronunciations. The other variant, within which São Paulo is placed, is less varied: it is either [ˈkaɾta], as in São Paulo city - and then in Paraná, Santa Catarina, Rio Grande do Sul, the Center-West, and the Amazon - or it is a retroflex "r" like in English: ['kaɹta], as in most of São Paulo hinterland, and parts of southern Minas Gerais and Goiás.
So, it seems that the specific pronunciation of post-vocalic "r" must be explained within the context of the whole Southern, Center-Western, and Northern regions, not exclusively of the city of São Paulo. And then we have to take into account not only Italian, but also Spanish and Tupi-Guarani, and Portuguese as it was at the time of expanding Portuguese colonial presence in Brazil. And, as far as I know, the uvular pronunciation is an innovation from the 19th century in Portugal. So the main hypothesis would seem to be, variant B, which is spoken in São Paulo state, is more conservative regarding the pronunciation of post-vocalic "r", not that such pronunciation is due to a foreign influence.
The pronunciations of /t/ and /d/
While the palatalisation of /t/ and /d/ when followed by /i/ is less common in São Paulo than in Rio de Janeiro, it is hardly an exception; indeed, it seems to be the most widespread pronunciation. The pronunciation of /ti/ and /di/ is very difficult for Brazilians(1); the tongue slips "naturally" from /t/ to /i/, due to the proximity of the point of articulation, resulting in [ʧi]. The other option is to change the articulation of the /t/ into a dental, thus setting the articulation point of consonant and vowel farther apart, which is characteristic of some Northeastearn pronunciations. This is present in São Paulo, too, and is popularly attributed to "Italian influence" - but given the equally huge, and more recent, internal migration from the Northeast to São Paulo, it would be necessary to discuss the probability that the "influence" in the case is Northeastern, not, or not only, Italian. Moreover, such dental pronunciation is common in Portugal, too, though I am not sure when it superseded the previous [ti] (or [te], as in the case of final syllables), and to what extent, and in which regions, it precluded the palatalisation. So, again, I wouldn't exclude the possibility of mere conservatism regarding those pronunciations.
(1) What is discussed about /t/ in the following applies to /d/ the same.
Interference of the Italian vocalic system
Regarding the pronunciation of stressed /o/ before /m/ and /n/ syllable onsets, I think the idea that there is nasalisation in the rest of Brazil, but not in São Paulo is mistaken. The pronounce of /o/ in such positions is non-nasal in every Brazilian variant that I am aware of - and I fear the same is true of European variants either. The vowel that notably changes its pronunciation before /m/ and /n/ syllable onsets, when in a stressed position, is /a/, and even there it is hardly a nasalisation; rather the pronunciation is [ɐ] instead of [a]. And this is true of São Paulo, both city and state, no less than of Rio de Janeiro or the rest of Brazil; it is [ba.nɐ.na] in both São Paulo and Rio, not [ba.na.na] in São Paulo and [ba.'nã;na] in Rio.
What could be discussed here is the denasalisation of unstressed nasal vowels, such as /'ɔr.gu/ instead of /'ɔr.gɐ̃w/ for "órgão"; but I would say that this is rather an innovation of Portuguese Brazilian, more or less as a whole, than a phenomenon focused in São Paulo.
Conversely, I do not hear paulistas or paulistanos failing to reduce final vowels, unless for emphasis (/'bɛ.tu/ if you are mentioning Beto, but /'bɛ.to/ if you are calling him). As far as I know, such pronunciations are characteristic of rural Rio Grande do Sul, western Santa Catarina, and Paraná, and they seem either conservative or due to proximity to the Argentinian/Uruguayan borders, rather than influenced by Italian immigrants.
Hiatus versus diphthongs
A better argument seems to be the hiatus in words like "italiano" ([i.ta.li.ˈa.nu] instead of [i.ta ˈlja nu]. This, at least, is characteristic of São Paulo, not of the whole Southern region, and seems to be innovative, rather than conservative.
Retention of final /l/
This is an obviously conservative trait, and I am not sure that it is present in São Paulo at all. It is more characteristic of Rio Grande do Sul, and even there it seems to be quickly fading; I still say /a.'zuł/, and even /a.zu.li.brã.cu/, but to my daughter the difference is inaudible. An "actual l" such as in /mal/ instead of /maw/ or /mał/ is something that cannot be heard in Brazil, unless it is in some very small and isolated area (or, of course, if the following word starts with a vowel).
There are surprisingly few words that passed from Italian (or Venetian, Sardininian, Calabrian, Sicilian, etc.) through oral transmission, as opposed to literary imports. They are not always easy to distinguish, but if they have a "c" and are pronounced with an /s/, they have not been brought into Brazil by Italian immigrants. "Cicerone" is pronounced /si.se.'ro.ni/, even in São Paulo, so Brazilians read it and started pronouncing it according to Portuguese orthographic rules. "Ciao", conversely, is /ʧjaw/, and so it was either brought here by Italian immigrants (or imitated from Italian movies).
Most words that are actually due to Italian influence (via immigrants, not via Italian literature or high culture) are gastronomical terms. Pizza, brachola, risoto, ravióli, talharim, espaguete , etc. (And even here, we say /ʁi.'zo.tu/, not /ri.'zo.to/, so it is the Portuguese phonology that actually predominates over the Italian, not the other way round.) And then those words are all common throughout Brazil; none is specific of São Paulo.
Grammar Singular used for plural nouns
This is widespread in all Brazil. See for instance http://www.avozdapoesia.com.br/obras_ler.php?obra_id=4432 for a poem in popular Portuguese, full of instances of that ("Venho dos brêdo", "vivo naquelas mata", "os cumpanhêro", etc). And it is Northeastern, not paulista, and I would dare say absolutely virgin of any Italian influence.
Enclitic pronouns instead of proclitic ones
This seems to be generalised in Brasil. It is not like people say "me dá" in São Paulo but "dá-me" in Recife. And so it seems to be wholly autochtonous - possibly related to prosodic issues ("dá-me" is more difficult to pronounce if we don't delete the final "e" as the Portuguese do; conversely, "me dá" is less natural when pronounced /m'da/.
Gerunds for progressive constructions
As Jacinto says, this is a conservative trait.
If we look to a massive case of actual influence of a language upon another - for instance, the undeniable influence of Norman French over English, in the transition from Old English to Middle English - we will see that such influence is predominantly lexical. And enormous amount of French words were borrowed into English, from "beef" to "lieutenant". But the phonological impact seems to have been much more limited; the most impactful change in English phonology, the Great Vowel Shift, seems unrelated to French. And if there were grammatical impacts, they did not go in the way of substituting French grammar for Old English grammar; the change from Old English to Middle English was an enormous simplification of grammar, with the demise of cases, most of gender, most of verbal tenses, etc. The complex grammar of Old English was abandoned, but the complications of Old Norman French didn't replace it, and were instead lost as well.
And so, though this is only one example, I would dare say that the influences of any language over another one are mainly lexical, only marginally phonetic, and, in terms of grammar, if any, mostly destructive, not constructive.
Do we have other examples? The influence of Arabic on Iberian Romance, or Tupi-Guarani on Brazilian Portuguese, of Germanic languages over Vulgar Latin, perhaps? If so, they seem to point in the same direction. Portuguese phonology seems to remain basically Romanic, evolving of itself, not of out of contact to other languages. It gained phonemes, but not from Arabic, Sueve, Visigoth, or Tupi. Or, more modernly, from French or English. It gained words, lots of them, from all those languages, and Quimbundo, Nahuatl, Japanese, or Tagalog. But it seems to have imposed its phonology over all those words, rather than having its phonology altered to adapt to such imports. And it doesn't seem to have gained syntactical features from any such contacts.