The only place you might feel even remotely irritated in this way is at Palmyra. There are the usual postcard-sellers (young boys), men on motorbikes offering to guide you, trinket-sellers, and a family of camel-owners in the valley of the tombs who will ask for money if you want to photo their camel. But none of it is in any way a big deal. Oh, and we saw young local kids at Rasafa (at the site and round the café) who were glad of a few coins - you might call them beggars, but they looked like they could do with the help.
I found that Syrian people are fine with foreign visitors, pretty much mind their own business, but will help you if you are in need. Ideal. :)