Anna
Firstly, Welcome to the Chandoo.org Forums
Excel and all spreadsheets work on the pull as opposed to push system. That is cells can pull data from another cell, but the other cells cannot push the data to the target cell.
In your case Cell B1 can be changed so that it reacts appropriately when A1 changes
eg:
=IF(A1=1,"", Existing formula in B1)
or
=IF(A1=1,0, Existing formula in B1)
Your choice of the Space or 0 in the new formula in B1 will depend on what is happening after B1
You can also use VBA to actually delete B1 but I'm sure the above will suit your needs