Does repeated xoring of the (same) key K lower the entropy of K?
If I take a strong key K of length $n$ taken from a good source of randomness, it will have $n$ bits of entropy and will be independent of anything else (not derived from it).

If we later Xor it with a weak password K* for instance or something else we still have the same $n$ bits of entropy. 

Each bit is still uniformly random and independent of each other. Even if K* bits were biased and correlated.

It is easy to see with an example. If we Xor a random key with all zeros, we get the same; if some bits are 1, it's just a negation, and negating a uniformly random bit is still uniformly random. Also, correlated bits won't make a difference. If I negate every other bit, or follow some other pattern, it makes no difference. 

However if you XOR with something dependent on the key, all bets are off, and you could end up with 0 entropy, for instance, by Xoring with itself.