Trim affects the attitude of the aircraft in flight. To fly straight and level typically the aircraft would be in a slight nose-up attitude. If the aircraft had a good cg (slightly tail heavy on most types) you would not need much stab trim to acheive this. If the load-planning was screwed up and the aircraft is too nose/tail heavy, this would require more stab trim to get the aircraft level. More stab trim leads to more drag so will increase your burn-off. Hence the smart fellows at MDD (MD11) and Boeing (747-400) came up with the tail-tank as a means to correct the trim without using the stab trim (ie no drag penalty). Getting extra fuel capacity was a bonus.
To keep it level at different weights require different power settings, ie less thrust for xxx weight. Trim is not a factor.
There is also trim change due fuel burn off as the fuel cg shifts as the fuel is used up.. previously when doing manual loadsheets we would have a template that would mark out the cg curve.. interesting as the cg would move fore and aft as fuel was used.
On widebodies on long flights, initially it will be very minor changes in cg as the fuel in the centre tank is used first. As this is in the centre section it has very insignificant impact on centre of gravity (ie not much trim correction required) but once wing tanks start supplying fuel, the cg changes as the fuel tanks are quite spread out due to the sweep of the wings.