Can SQL Server WebAdmin degrade the performance of my site?
Any application that runs queries against a SQL Server (or any database engine) instance will decrease the performance because it's extra load the server has to deal with. I assume the warning is just to cover themselves, legally.

What it comes down to is the scripts you run, and how much activity the application generates on its own. There may be options to disable the latter, which is usually a good idea when working against a production instance.

I'm not familiar with the particular product you mentioned (Azure?), so for example from Management Studio, I always disable IntelliSense by default on a production management box, because this runs queries behind the scenes to do its work. Activity Monitor is also a big consumer of resources, and sometimes people leave that tab open and forget about it.

So again, their warning is a standard thing; I would probably just ignore it.