Barrier contraceptives (condoms, sponges) will not affect breastfeeding.
Estrogen containing birth control pills will decrease milk production significantly in the majority of women and are not recommended for nursing mothers.
The progesterone only oral contraceptive (mini pill) is compatible with breastfeeding and will not affect milk production in the majority of women. However, rarely, I have had patients whose milk supply did decrease significantly after starting the progesterone only pill.
IUDs which release progesterone should not affect milk production. However, before having an IUD inserted, you may want to use the progesterone only pill to see if it affects your supply. If so, reconsider the decision to have the IUD inserted.