More weight decreases range. The XX Air Force had problems with planes running out of fuel flying from Siapan, Tinian and Guam. Reducing the bomb load helped extend the range along with some other changes like flying at a lower altitude (not flying into a head wind such as the jet stream helps too).
The effects are not that large if you do not have to reduce the fuel load in order to carry the extra bombs. IIrc, the B-17 manuals gave weight dependent ranges in 5000lbs increments, so basically, there are cases where it would not matter if you took off with 5000 lbs extra bombs. On the paper. Maybe someone has the pages ready.
A rough estimate would be 10% extra weight to equal about 10% extra power at cruising speed, meaning about 10% less range but then there are many variables.
There are a number of different tradeoffs, depending upon the aircraft; a heavier internal load might only affect climb, but if your bomb bay was also the location of your extra fuel cells, a greater sacrifice of range might be traded for a bigger punch.
Some aircraft included the option of loading bombs externally, and depending upon the type and number of external stores, the drag could be an issue as well. A lot of prewar designs had internal bomb bays that could not hold what wartime experience proved to be a 'useful' load.
- longer take-off runs (may lead to limitations)
- lower climb-rate
- higher power-setting during climb or considerably longer time to altitude (both give fuel penalies)
- higher drag in cruise
- higher altitudes may not be accessible