It can, but I don't think you're going to like the reason why. Mattresses typically accumulate dirt, human skin, and bugs as they "go through life". They hopefully won't change the weight significantly. Otherwise, you need to check your cleaning schedule.
Mattress weight does not change over time. It may sag over time and look different, but unless you actually remove or add material it will not change weight.
Mattress weight may go down over time as the particles and products used to make the mattress break down, however this will take a good bit of time. The mattress may also appear heavier as the insides shift and change position with age. Time has a big effect both ways.
It can, yes, for the very gross reasons that you might suspect. Mattresses accumulate flecks of human skin and dust mites, and any absorbed fluids will also add weight.