In short, for humans who are not premature babies, the answer is “no“. Not in the traditional sense of taking in oxygen for the rest of your body, anyway.
Your skin’s essential function is to prevent things from the environment from entering your body, and things you need (e.g. water) from leaving the body. It’s very good at its job! Skin is also necessary for secreting sweat to cool the body down. However, these are the only two really essential functions of skin. Other functions are pretty useful though, such as touch and making vitamin D from sunlight!
The idea that skin has a role in breathing, and that you can suffocate if your pores are covered for too long, probably comes from the James Bond film Goldfinger, in which the villain kills his victim by covering her in a thin layer of gold. People at the time (1964) believed this, but we know now that it isn’t true!
However, the very top layer of your skin (0.25-0.40 mm) does get almost all its oxygen from the air – not through the pores, but through direct diffusion (imagine that the oxygen is soaking through into the cells, like water through a piece of cloth). However, your skin doesn’t need to be in direct contact with air for oxygen to diffuse in – oxygen dissolves in oily substances like makeup and moisturiser, so it can diffuse through the whole lot and get to the skin, even if you manage to cover it 100% (which is quite difficult).
Still, it’s a good idea to remove your makeup every night before you go to sleep – pores clogging with the day’s dirt and oxidising makeup won’t suffocate you, but it can lead to acne and blackheads.
Interestingly, some animals, such as frogs, do get most of their oxygen needs through their skin. So don’t go dipping your pet frog in gold!
M Stücker, A Struk, P Altmeyer, M Herde, H Baumgärtl and D W Lübbers. The cutaneous uptake of atmospheric oxygen contributes significantly to the oxygen supply of human dermis and epidermis. J Physiol 2002, 538, 985.
D F Roe, B L Gibbins and D A Ladizinsky. Topical Dissolved Oxygen Penetrates Skin: Model and Method. J Surg Res 2010, 159, e29.