Is there a strategy PVP board/card game with Negative Feedback loops?
The great granddaddy of deck-building actually does have a negative feedback loop in it:

In Dominion, victory points come in the form of green cards which you add to your deck[1]. In every game of Dominion you will have access to three basic Victory cards - Estate, Duchy and Province - and there are others that get added as part of choosing the kingdom for a given game. The basic Victory cards, and many of the others, have no playable effect, meaning that a player who jumps straight into buying them gets an initial boost in their score, but will soon find that their hands are full of junk cards that they can't use to progress the game. By comparison, a player who focuses on building up a strong engine in their deck may not have any meaningful score until the 10th turn, at which point they can buy multiple Provinces to end the game and win.

Another game which has a kind of negative feedback loop is Power Grid. In Power Grid, the turn order is determined based on increasing number of cities. If you have more cities (and can power them), you get more money, but you also get last pick when buying resources and building new cities. This can mean you get screwed out of both, putting you at a severe disadvantage. This effect is actually so strong that positioning yourself correctly in turn order is an important part of the strategy of the game, and some people dislike it because it becomes such a major component of the game.

[1] In a couple of the expansions, namely Prosperity and Empires, there are ways of acquiring victory point tokens which do not clog up your deck. And in many of the expansions, there are Victory cards that can also be played as Actions or Treasures, so they're not completely useless during the game.