No, you don't. We can understand why you would want your legal designation included, e.g. LLC, Inc., LLP or Corp., but it's not required - even if your lawyer says it is!
Can we add it into your logo anyway? Yes, of course. Your call. Before you decide, see if you can name a few large companies that include a legal designation in their logo. Nike? No. Facebook? No. Google, Sony, Twitter, Adidas, Visa, Ferrari and Coca-Cola? No on all counts. Not one. This is because they use the logo as a company identifier. Almost like a signature for the company. When you see that swoosh, you know it's Nike. Adding the legal designation adds clutter, which diminishes impact. The cleaner and simpler your logo is, the better.
Those legal abbreviations belong on legal documents, disclaimers, letterheads, etc., not on your logo.