I am wondering what the problems would be for a newly married couple that do not have any children to use birth control and what are the sources for these problems?
The first mitzva that is written in the torah is pru urvu- to procreate, which is one of the positive commandments that every Jewish man is obligated to do. In fact one of the main reasons we are commanded to marry is in order to perform this mitzva. Additionally we are not allowed to stall off performing this commandment. The gemora Nidda 13 (Yesterday’s daf yomi) says that pushing off having children delays the coming of moshiach. The gemora there is talking there about marrying a young girl who is not yet of childbearing age, because the man is then stalling off performing this mitzva. That is talking about a person putting himself in a situation where he can’t perform the mitzva, taking birth control is worse because the person is actively pushing off doing this great mitzva. (See Chelkas Yoav E:H 60). The Rambam (Ishus 15-1) writes that a man is obligated not to miss any opportunity to perform pru urvu, (such as stalling off the mikva) until he has performed it. (Also see Y:D 197-2 and Taz 197-2). Even after the couple has a girl and a boy there is still the mitzva d’rabonon of “sheves”- to inhabit the world with more people, therefore taking birth control even after the couple has performed pru urvu is problematic, but not as severe as beforehand.
Having said that, there are times when a couple will have halachic permission to take birth control, however it should be done with the guidance of a competent halachic authority.