There are extensive answers on our sister site, Science Fiction & Fantasy, addressing this precise question.
Rather than copy the full details I will copy the relevant points here and link the full answers.
In fact it seems likely that there WAS some ice on other trips but due to the ambient temperature and humidity this may have not been apparent or melted before we see it.
The formation of ice requires at least 3 things.
- (The normal, linear passage of) Time
The time factor is clearly seen in both Einstein's short trip and Marty's trip from 1955 back to 1985 at the end of the movie.
Here is the return of the DeLorean with Einstein just as the car is spinning to a halt.
But here it is a few moments later.
Here is the car just moments after it has crashed on return from 1955 to 1985. Some mist, but not much ice, if any.
And by the time the terrorists are coming to get revenge on the Doc., ice is apparent.
So the Doc. mentioned cold, we've seen the effects of time, the other factor is moisture, or in this case, humidity.
The scenes where Marty goes from 1985 to 1955 are slightly different, in that the car is surrounded by mist, but there is little evidence of ice on it while in the barn. By the time Marty drives it back out of the barn, the effect has vanished.
Here is the first part of the time in the barn, the mist is obvious but there is not much ice, if any.
A few moments later we see that the mist has cleared, and though there is no ice on the car, there is a sheen of moisture on the bonnet and windscreen.
My conclusion is that in this trip, it was a lot less humid (to gain less ice), & significantly warmer, to melt it faster.
Source answer by Andrew Thompson
Out of Universe
The out-of-universe reason for the disappearing ice was explained in the commentary track to the first movie (transcript here) with writer Bob Gale and producer Neil Canton, and Gale also suggested the in-universe reason could have something to do with the switch from the plutonium power generator to Mr. Fusion:
Bob Gale: I believe this was liquid nitrogen that we had sprayed all
over the car to ice it up.
Neil Canton: It was also a problem keeping it on the car... it would
drip off several times before we were ready to go.
Bob Gale: This is the iciest you'll ever see the DeLorean. You'll
notice as the movie goes on, when the DeLorean reappears from a time
trip, there's less and less ice on it, and finally by the time we're
into the sequels there basically isn't any. I figured we used Mr.
Fusion as an excuse why we weren't gonna have ice on the car anymore -
great idea, great visual, and a pain in the ass to shoot.
Source answer by Hynosifl