Such a word illustrates the façades and masks that mankind dons in order to hide their true emotions.
You could also use: "...and masks, which mankind dons...". I'm not sure how well "their" holds up in the end of the sentence. Maybe "its" would be more appropriate. I'll leave that for someone else.
If you want to treat mankind as a singular noun (ie 'dons'), then you need to match it with 'its' at the end of the sentence. It's definitely less conventional to treat mankind as a plural noun, kind of as a substitute for 'all men', but - if you do - then you can keep 'their'. In this context I would absolutely go for the latter, actually: because mankind doesn't collectively don one single mask to hide its one single emotion; instead, each individual person composing mankind does. So the slightly less conventional plural noun is both more striking and more logical.
The faint, stale smells of beer from the previous night suggest that humanity has returned to methods of denial and forgetfulness, as a way of evading their sordid realities.
You needed the comma between faint and stale. "Smells" is plural, and thus, it should be suggest. Furthermore, humanity is singular (although referring to a plural, that fact is irrelevant) and thus it should be "has" instead of "have".
Leaving the comma out between 'faint' and 'stale' is technically incorrect - but it could be a stylistic choice, and if it's a quotation you can't change it.
In terms of the 'that' versus 'which', the more correct word would absolutely be 'that' (as corrected by the above poster) because what follows is a definitive or restrictive clause.
Lastly!! In terms of the plural versus singular: I believe it should be a singular here, because you're not saying that the *smells* suggest... you're saying that the quotation as a unified, singular object suggests it.
Totally agree on 'humanity has'.