No, not for safe landing. If a jet couldn't land safely while fully tanked then they would not be able to land in emergencies happening early in the flight.

They rely on the plane being lighter from fuel burn and therefore more efficient to extend their possible flight time, but not for safety.

I guess it depends on what you call a safe landing. For large airplanes the maximum takeoff weight is typically greater than the maximum landing weight. If an airplane lands overweight it can no longer fly until it has been inspected and repaired. Most emergencies will involve flying around for long enough to get the aircraft within the acceptable weight limit. Some older aircraft have the ability to dump fuel, although I don't know of any new ones that have this capability.

Needing to land before getting under weight usually means a crash landing. In modern times, very few things like fire or an engine self-destructing will require immediately getting out of the air. It is usually better to fly in a straight line, figure out what is going on, figure out where to land, then land. This lets you get underweight, lets you communicate with air traffic control, lets emergency services prepare, etc.

It isn't an issue for smaller planes (A320, 737), but larger planes (such as 747, 777, A340, A380) usually have a lower maximum landing weight than their takeoff weight. If an emergency happens early in flight they usually have to dump fuel to reduce weight before they can land. https://en.wikipedia.org/wiki/Fuel_dumping

An emergency overweight/crash landing is the opposite of a safe landing.

Airliners will dump fuel or keep flying to get below maximum landing weight if possible. If they're still overweight and manage to land without issue, the plane will need to be inspected and repaired before it can fly again.

Isn't fuel vapor more dangerous than liquid fuel though?

Contained vapor creates an explosion, while liquid must evaporate first.

The strain the weight of the fuel places on the airframe during landing is the issue, not a risk of explosion. Furthermore fuel tank inerting systems can greatly reduce the likelihood of any problems.

Sure, but I'm not sure what you're asking in context. It's not about fuel vapor but overall weight on the airframe at landing. The bigger jets are not designed to touchdown at the fully loaded weight.