Re: bearing in mind that my Thor knowledge is limited to Simonson and some Avengers...
Probably the worst possible effect (potentially, depending on the writer) of the Lasso around Thor would be 'revealing the truth' and reverting him to Donald Blake. We could go 'round and 'round debating what his true form is (personally I think of him as Thor = true, DB = the disguise), but it would come down to who wrote the story. I think you could also have a 'clash of magic' in that event, with the Lasso's magic in direct conflict with Mjolnr's.