Firstly, wow, thank you for letting me know about IUDs. I'm sure I could have found this out with 5 minutes of Googling, but you saved me the trouble, haha.
Anyways, hormone-based birth control will affect weight loss, at least with estrogen-based BC. I imagine a lot of weight that women carry on BC is water based because estrogen increases water retention. However, it is also known that the test:estrogen ratio is important in determining muscle anabolism. Estrogen based BC floods a woman with estrogen, and this I would imagine would shift her T:E such that this type of anabolism is diminished. What this means is, the muscle would be less metabolically active, so she would burn less calories in a steady-state scenario. So in essence I think it would make weight loss harder, yes.
However, it seems like at least non-hormonal IUDs could negate this primarily by not being hormonal. In addition, I still haven't done the research yet, but I suspect that progestin-based BC would also not affect weight gain negatively.
Subbing for feedback from anyone else.