Certain dyes can be quite harsh to the scalp, and can irritate open lesions, but if milder dyes are used, patch tests carried out, and reputable hairdressers consulted, there is no reason why someone with psoriasis should not dye their hair. This also applies to the use of other hair products.
The face is a delicate area, care should be taken that makeup does not irritate it, and it is always unwise to apply cosmetics to open sores. However, like with hair products, if a patch test is carried out and no irritation occurs, there is no harm in applying makeup. This is the same for fake tan on the body.
Remember, however, that tanning beds have health risks, which can be increased if the individual is undergoing UV light therapy. Fake tanning products such as bronzers, gels and lotions are the safest way to tan. You might also be interested in finding out about camouflage, specifically designed to cover scars and skin conditions such as psoriasis.