You sure its the not spending money that is awkward?
Accepting an invitation to dinner with a group of people and then refusing the eat . . . yeah that's awkward.
My BIL's diet used to be eat once a day and he didn't drink; he'd routinely go to dinner with friends, family, etc. and not have anything but a glass of water. Once the food came it would get weird.
Then there is Penn42's point, it is generally not socially acceptable to go to a location that offers primarily a service (as opposed to shopping a store w/o buying) without a cover change and not partaking. But, unless all of the tables are full and you attend moves the families table to a larger size (say 4 top to whatever the next size is) I am not sure I would go as far as to say you are costing them money by displacing a paying customer.
Try not accepting the invitation. Try offering the invitation first (to a place you like or to have them over). Try telling a white lie (saying for a down payment, trying to lose weight, food sensitivity, doctor's orders, already have plans).
Of course money is awkward especially among friends. And heck it is a way to show off.
For example, you stay in with your buddies and drink at your place and you are buying drinking (albeit at a much cheaper price for all of them). If they try and pay you it would generally be awkward (so why not pay a bartender more), if they bring cheap beer and you offer pricey 15 year single malt it would be awkward, if they never host it could be awkward, and so on. Not saying it should be, but it often is.