The short answer first
Are there significant reasons for a pianist to know the key in which a piece is written?
Yes.
If so, what are those reasons?
Music theory is not difficult if explained well, in context of the piece you are playing. Otherwise, it's boring like doing senseless sums. My suggestion is to start by learning theory that pertains to pieces that you are interested in.
Pieces and musical ideas are not pulled out form a void. They contain some essence and meaning.
Any infomation about the composition helps you understand it. Background information is important and so is how the piece is constructed. Theory will give you an idea of how the piece was constructed.
This will help you play better.
So lets start with some theory... A crash course in the theory (or history?) of key signatures.
Most compositions, unless you play an atonal (12 tone [13 tone if you include the octave

] , eg. schoeenberg, berg, etc work) work has a key center. This simply is a set of notes (amongst all 12 tones) that are more commonly used than others. For historical and some (perhaps vague) mathematical reasons, we usually have 8 tones in a key. The eight tones can be "derived" from how you divide up a piece of string in equal parts - 2, 3, 4 sections.
The number of sounds that you can make by dividing the piece of string up follows a fibbonacci series... 1 2 3 5 8 13 (this is because some of the ratios are degenerate e.g. 2:4 is congruent to 1:2).
The common 5 tone scale is called the pentatonic scale. Oriental music is based mostly on this scale. Unlike a string instrument, with keyboards, it is not possible to have microtones. Early keyboards used a 13 tone system over an octave, where 8 of which were used for the particular key. However, as you expand the system over several octaves, there were several problems with having a simple 13 tone system. Bach partially solved this problem by popularising the well tempered scale. (Therefore the well-tempered clavier.) The thing about this systems is that each key had a different quality associated with it.
In the late 19 century (or there abouts), a different system called equal temperment became popular. Here all the keys had the same quality about them. This is the system we use today.
The two (three) 8 tone scales that are common in western music today are the major and minor scales (two of them).
Now you know about keys. No problem!
al.