I saw Dan's answer. In the UK, some bars still run "tabs" in this way. In a more formal bar, they would probably want your credit card first in case you disappeared without paying. In a bar for local people, the tab could be more informal, based on trust built up with regular clients and a new client shouldn't expect that he/she could open a tab. When you order a drink and you have a tab, you say "put it on the tab". This expression can also be used in a funny way between friends. If I ask a friend for several favours, he could suggest after the first one that he will "put it on the tab" i.e. expect payment later. "Tabs" are also used on web browsers. When you want to access a new internet site but keep the existing one available, you open a new tab.

I don't think businesses do this any more, now that there are credit cards. A century ago, some businesses, notably bars, would establish credit accounts for customers. "Tab" can mean such an account. "Can I open a tab?" could mean "Do I need to pay you right now? Or can you just keep a running total of what I spend here, and send me a bill at the end of the month?"