Is there a cost to having many different addresses? Whenever I want to receive funds to my bitcoin wallet, it gives me a different address and says that it is "your current bitcoin address". 
Suppose I collect money from 100 different people, and then want to spend all this money in a single transaction. Will I have to create a transaction with 100 different sources (the 100 different addresses) and a single destination? Will this transaction cost more than a simple one-source one-destination transaction (e.g. because it is larger)?
Will I have to create a transaction with 100 different sources (the 100 different addresses) and a single destination?


Yes.


  Will this transaction cost more than a simple one-source one-destination transaction (e.g. because it is larger)?


Yes.

However, this has nothing to do with using multiple addressses or not. Whenever you receive money in 100 transactions, it creates 100 unspent transaction outputs (coins). If you want to spend them, you need to redeem all those 100 coins. Whether they're all sent to the same address or not is completely irrelevant.

So:


  Is there a cost to having many different addresses?


No.