You and your tricky questions! This is a tough one. Ok, in European Portuguese you have two models: esquecer (something/someone) and esquecer-se de (something/someone); in Brazilian Portuguese you have these and also esquecer de (something), (see this question). I think esquecer de is equivalent to esquecer-se de, but I’m not sure, so I’ll write about usage in Portugal only. In many, but not all, cases context determines the meaning regardless of the model you use. But generally you’re more likely to use esquecer-se de when something slips your mind; and just esquecer when you disregard or deliberately ignore something, or when you no longer think of something because you moved on, lost interest, even though you could recall it if you wanted. Here are some examples; remember, in many cases you’re just more likely to use one form rather than the other:
Esquecer-se de (‘slips your mind’)
(a) Esqueci-me do teu número de telefone
(b) Esqueci-me das horas, e quando reparei já era meia noite
(c) Esqueci-me da reunião [I forgot I had that meeting]
(d) Desculpa, esqueci-me de ti [I forgot you were waiting for me, or something like that]
(e) Esqueci-me de regar as plantas
The other option, just esquecer, is less usual but ok in (a) and (b), uncommon in (c) (people might think you decided not to attend the meeting) and (d), and impossible in (e); this last one may have something to do with the fact that the thing you forgot is expressed by a verb (regar) rather than a noun, but it is nonetheless true that the interpretation is ‘it slipped your mind’.
Esquecer (‘disregard, deliberately ignore, no longer think of because you moved on’)
(f) Esquece o passado e vive o presente
(g) Esquece essa mulher; ela não quer nada contigo
(h) O João ganhou a lotaria e esqueceu os velhos amigos
(i) O presidente esqueceu o protocolo e aproximou-se da multidão
(j) Depois das eleições, os políticos esquecem as promessas que fizeram
(k) Foste desagradável, mas eu já esqueci isso há muito tempo
(l) Nunca esqueceu o seu primeiro amor
Esquecer-se de would be unusual in (f) and (g); in (h) it would more likely be interpreted as ‘he didn’t share any of the prize money with them’; in (i) to (l) the literal meaning would switch to ‘slipped their mind’, but it might be interpreted as figurative speech and make no difference. For instance, you could say os políticos esquecem-se das promessas que fizeram. But it is very unlikely that a politician will no longer have a recollection of the promises they’ve made. So people would likely think you’re just being ironic and real mean they no longer care about those promises even though they remember they’ve made them. Same with (i): it is more likely that the president disregards the protocol rather than that he really does not remember it. So o president esqueceu-se do protocolo would probably be taken as jocular, and possibly somewhat appreciative, way of saying the president disregarded the protocol. In (k) if I say já me esqueci disso (patently it has not slipped my mind, or I couldn’t talk about it), I’m just speaking figuratively (I no longer think or care about it, it is as if I no longer had any recollection of it). That’s why I say that in many contexts context rather than the model will determine the meaning.
Esquecer(-se de) can also mean ‘forget (something somewhere)’. You can use both models:
(m) Esqueci as chaves em casa.
(n) Esqueci-me das chaves em casa.
Reflexive and Inherent se
I’d say of all instances of esquecer-se you come across 99.99% are not reflexive. None of my examples above are. (p) below is a reflexive example:
(o) Eu vesti a criança
(p) Eu vesti-me
The verb vestir has the same meaning in both (o) and (p): the only difference is that in (o) I perform the action on the child; in (p) on myself. In examples (a) to (e) and (n) I don’t perform any action on myself. That me is an inherent pronoun, i.e. it is part of the verb: you should think of the verb as esquecer-se rather than just esquecer. One way to tell the difference is that in a reflexive verb you can reinforce the pronoun with a mim mesmo, a ti mesmo, etc., but you cannot do that with inherent pronouns:
[Fine] (q) Eu vesti-me a mim mesmo
[Wrong] (r) *Eu esqueci-me a mim mesmo das chaves em casa
There are quite a few verbs with the inherent se in Portuguese. Celso Cunha and Lindley Cintra (Nova Gramática do Português Contemporâneo, Lisboa, 2014, p. 512) list a few:
[…] admirar-se, apaixonar-se, arrepender-se, atrever-se, casar-se, enganar-se, esforçar-se, esquecer-se, indignar-se, lembrar-se, orgulhar-se, queixar-se, rir-se, sentir-se, suicidar-se, zangar-se.
Ok, what if you really want to say I forget about myself (and only think about the others’ welfare). In theory you could say Eu esqueço-me. The problem is people would most likely interpret this as the esquecer-se de model and ask “esqueces-te de quê?” One solution is to use the esquecer-se de model:
(s) Eu esqueço-me de mim (mesmo)
[For comparison] (t) Eu esqueço-me das minhas necessidades
If you want to use the straight esquecer model, then reinforce it with a mim mesmo:
(u) Eu esqueço-me a mim mesmo
[For comparison] (v) Eu esqueço as minhas necessidades
I must say I don’t think I’d ever come across anything like esqueço-me a mim mesmo. But I looked around in the web and found a few examples.