I have a MTB which has a 1.600g frame and 2.600g suspension fork. Does being heavy at the front cause a disadvantage when climbing?
Weight anywhere makes climbing harder.
Weight over the front end is somewhat beneficial when on a steep climb, because it helps give the front wheel traction, which helps with steering and balancing.
If your front wheel is not carrying enough weight, then turning inputs are slower and eventually ignored completely.
Being tall, I find steep climbs a challenge because significant grades put more weight behind the rear axle, resulting in a "lifty" ride. Minor changes like moving my full waterbottle to the front cage can have a surprising impact on this.
Do remember that the rider will outweigh the fork, frame, and the entire bike by a wide margin, so your body posture is the single biggest effect on weighting of the bike.
Bike frames are designed for a certain geometry, so swapping out a fork for something random might raise/lower the front of the bike to a point where it really doesn't 'play nice' with how the frame is designed. I would stick to a similar fork in terms of height/offset/etc as the original frame spec.
In terms of weight, some weight at the front when climbing might actually be helpful to keep the front wheel down and give you more traction. However, as soon as you get into more technical climbing where you're lifting your front wheel, doing j-hops, etc you might find the extra weight starts getting in the way!