I would argue that to an extent, there is a sort of psychological effect, but it depends on the situation.
For example, a player rated 1500 might be in a tricky, but winning endgame against a player rated 2000. 2000 might very well offer a draw, even though he knows he is losing, because a) 1500 might balk at the rating difference and accept or b) 1500 might play on and get tangled in some sort of complex mess. For 2000, it is a win-win situation, if he is really losing.
Now if the ratings were flipped and 1500 was losing and offering the draw, many a 2000 player might feel rather insulted, or at the very least, annoyed. This is where I think psychology may play a bigger role. Unfortunately, it involves some rather questionable issues of ethics and poor manners.
Here's how I handle the draw. I generally don't like offering the draw. If I'm winning, offering seems like a concession, no matter how high the opponent. If it's even, my opponent can always offer or we can play it out. If I'm losing, offering will either annoy my opponent or show a sign of weakness. I generally accept draws when I feel that I am losing or the position is even and relatively obviously so (meaning that there is no play left). I have played many a game against higher-rated players where I could probably have gotten a draw had I offered in a winning position but instead lost the game. This is not to be blamed on the draw, but in my lack of technique. So I must learn from those mistakes.
Is it just me, or have you noticed when you offer a draw from a roughly even position, a lot of times if your opponent does not accept your draw it will cause them to have some sort of meltdown where they over extend trying to attack you leaving themself vulnerable?
Here is an example from one of my games