忍不住 ，忍不了 ，受不住，受不了seem to be fairly synonymous in Mandarin, but there does seem to be some subtle differences between them.What exactly is the difference? When should I use each one? Could you give me an example sentence to show when each of these terms would be the most appropriate?
First of all, 忍 emphasises inwardly endure, the condition can be either external or internal, 受 emphasises passively bear/accept, the condition is generally external.
You can say 我 忍不住 笑了, but not 我 受不住 笑了. 忍 could be paired with a neutral condition such as curiosity, but 受 generally only pairs with perceived negative conditions.
忍不住 -> 'can't hold/can't help/can't withstand but to + [action]'. It is often used in connection to an action to establish a cause and consequence logic. The subject make an action because they can't withstand a certain condition. The condition can be mentioned in the context from before or it could also be explicitly identified.
我 (I) 忍不住 (can't help) 笑了 （but to laugh)
我 (I) 忍不住 (can't stand) 热 (the heat), 只好 (have to) 出去 (go out)
受不住 -> Roughly the same as 忍不住, but generally apply to can't withstand a physical condition. Personally I rarely see it being used.
忍不了/受不了 -> 'can't handle it anymore/can't stand it anymore/feed up with it/can't cope with it'. These expressions carry intense emotion, commonly anger or frustration.
我 (I) 忍不了 (can't stand) 这 (this) 人 (person)
我 (I) 受不了 (can't handle it anymore) 了！
As Scharlach Nachtfalter has pointed out,
忍refers to internal endurance, whereas
受refers to reception of something (usually external and negative). In fact, when we say
忍受, it is a combination of the two – you receive something from the outside and then you endure it in the inside. However, in the case of enduring something from within (e.g. containing oneself from laughing), you can only use
～不了 (liǎo)is not pronounced as the usual
le. This is because it is originally a verb meaning 'to end; to finish' in classical Chinese. Consider the example
這醋太酸，我喝不了('This vinegar is too sour, I cannot drink it'): if you parse it very literally, it means 'to not finish drinking'. But gradually this became a way of stating potentiality ('cannot drink') in modern Chinese. (Think of that as, if you fail to finish doing something, that means you cannot do it.)
忍不了incline to mean 'cannot tolerate' and 'cannot contain oneself'. It is therefore natural to put this verbal phrase before a grammatical object (because
受/忍are transitive), as in
忍不了這個人. The main verb of the sentence is precisely
～不住means 'to stop' in classical Chinese:
From both sides of the gorge come incessant howling of gibbons and monkeys (translation)
受are deliberate efforts to halt the external / internal influence. If you cannot 'stop' them, as in
受不住, your efforts are nullified. But your efforts on what?
You therefore match it with another verb, as in
哭, 'cry'). The focus of the sentence rests in the verbs that follow, not
～不了, it does not indicate potentiality (e.g., there is no such thing as
*喝不住, cf. point 2. first para.).
All of the following sentences have something omitted (especially so in speech).
Sentence i. is the most idiomatic: the disturbance obviously is extrinsic, and the thing that is omitted at the end of the sentence is obviously the act of speaking on the phone loudly (a nominalised phrase).
Sentence ii. is also idiomatic, but its meaning is different. You cannot withhold something inside – probably your discontent as a result of the disturbance.
Sentence iii. predictably has an omitted verb after
不住. But 'talking on the phone' does not act directly upon yourself (you are not the intended object of the verb). There is no possible verb candidate here. So this sentence is nonsensical, perhaps ungrammatical.
Sentence iv. is perhaps idiomatic, because you can be suppressing the urge to scold that person ('to scold' being the verb).