Wait a minute
This is pretty generic and informal and is often used idiomatically to ask someone to pause what they're doing until told to continue. It doesn't usually mean exactly a minute, any more than "wait a second" means exactly a second.
Wait for a minute
This sounds like something that you'd find in directions/instructions:
Wait for a minute and then add the oil to the pan.
In this case, it more often means exactly a minute, though it's not required. The exact timing here is more important. You would be unlikely to find native speakers who would use this in the idiomatic manner of the previous example.
Now, does this carry over to numbers of minutes greater than one? Sort of?
Wait five minutes
Will usually mean five actual minutes... otherwise, why bother specifying the number. As mentioned before, "wait a minute" doesn't really mean a minute... one could end up waiting for a few seconds or several minutes in the end. Even in directions, you'll often find they've left out the "for" in this case:
"Remove from stove after one minute of boiling and wait five minutes before topping with granola."
So what about with the "for"?
Wait for five minutes
This isn't wrong. Using it this way is not ungrammatical at all, it's your choice. It's not required and you will definitely find examples of it, particularly in more professionally-written instructions:
Slide the whole skillet under the broiler, and wait for five minutes, until the crumbs are golden brown and the salmon is done.
This is also discussed, though not to a great degree, in Cambridge Dictionaries Online:
Wait means ‘stay in the same place or not do something until something else happens’. We can use it with or without for:
- Put a tea bag into the cup, then add water and wait (for) a minute or two before taking it out.
- I phoned the head office but I had to wait (for) five minutes before I spoke to anyone.
