TL;DR: Large aircraft normally isolate the left and right fuel systems, so if engines start, you have enough accessible fuel to take off and divert or return. There is also a cross-feed mechanism to rebalance fuel in flight, so most failures can be corrected.
Generally in small GA aircraft one of the first things you do on the ground is switch fuel tanks to ensure proper flow from both.
Small GA aircraft have one engine and two wing tanks, so there is a selector valve and potential latent faults, e.g. a blockage affecting one tank. A jet transport, like a 767 has at least two engines and at least two wing tanks. In normal configuration, each engine solely feeds from its own wing tank, so if the engines start and run, it is a sign that each side is functioning properly.
Aircraft also have a cross-feed system, which is not normally used, but is important during an engine failure. In particular, a stuck-open crossfeed valve may cause fuel to preferentially feed from one side, leading to an imbalance. This isn't an uncorrectable problem, as you can use the cross-feed to rebalance the fuel via switching on and off fuel pumps, but a return is likely warranted.
Where the isolation breaks down is the center tank(s). Here, you have one center tank and separate pumps to transfer the fuel to each engine. Potentially, one of the center tank pumps failed, so only one engine could be fed from the center tank, which would eventually cause an imbalance. This could be problematic if the cross-feed valves were to fail.
Airbus in particular inhibits center tank feeding on takeoff for isolation purposes, so you can take off and then realize your center tank fuel is unusable. Boeing does let you feed from the center tank at takeoff. In both cases, this isn't a safety issue, as you burn center tank fuel first, and you have more than enough wing tank fuel for a return or diversion.
Another issue is fuel leaks. The fuel system consists of a complex series of piping, motor-operated valves, and check valves. Some failure modes can cause fuel to flow in an incorrect direction. Fuel piping is, to the greatest extent possible, contained inside tanks, so a leak won't result in a hazardous spill, but may manifest as a unwanted fuel transfer.