When the aircraft is going to be overweight, then yes, the airline can only take what weight the performance numbers will support. That's somewhat uncommon, but it can defintely happen when takeoff performance is limited (hot day, short runway, full airplane, etc) or when the landing weight will be at the maximum allowed landing weight (full airplane + significant fuel load required at landing due to requirements for an alternate airport, for instance). (More on this topic and how the numbers work in this answer.)
As far as a passenger being overweight, that's less likely to be an issue, simply because most airlines (in the US, certainly -- this may apply less in other countries) use an average weight multiplied by the number of passengers, rather than weighing each passenger individually. While the difference between four ballerinas and four linebackers in a C-172 is very significant, that same difference among 100+ passengers on a commercial airliner just evens out over everybody else onboard. So there isn't really a case where the airline would say, we can't take this passenger (weighing 300#) but we can take that passenger (weighing 100#), because we only have 250# of weight left. Either you can take one more passenger, or you can't. Doesn't matter what he/she weighs.