Elopement Dinner

If you’re formally inviting people, you are responsible for the bill. If it’s a casual “hey, we’re going to dinner at xyz if you want to join,” you aren’t necessarily responsible, but I think it would be nice to offer.

If the dinner is the only reception you're having immediately after the ceremony, you need to pay for it. The reception is your thank you to your guests for attending. On the other hand, if you have at least a cake and punch reception afterward, you can do the “hey, we’re going to dinner at xyz if you want to join” thing afterward.

As for alcohol, it really depends. People do have dry weddings (so long as nonalcoholic beverages are provided). But only you can know whether your close family and friends would normally have wine with dinner, for example. If they would, you might at least want to provide that, even if you didn't do an open bar.

You should try to host your guests who attend your elopement. Check out what various restaurants have to offer, a lot of times you can customize the menu including drinks. You could probably ask for a limited beer and wine menu, and since you're are just doing dinner, most people are probably going to have between 1 and 3 drinks over the course of dinner. Depending on how many people you have, it should still be pretty manageable. Look into BBQ, Mexican, and Italian places, you generally will get more bang for your buck with those.

You should foot the bill. Depending on how many people are attending, it may be cheaper to cater a nice meal.

I'd pay for it if you're formally inviting people.