If a mother took tetracycline during pregnancy, would the infant's baby teeth come out stained - or would they emerge white and stain over time?

Come out. The stain would be on the teeth as they emerge. It could be just on the primary teeth or also on the permanent teeth depending on what was forming while the tetracycline was taken.
Usually Tetracycline. Given in 2nd or 3rd trimester deposits in "baby" teeth. Given near term, it can deposit in the crowns of permanent teeth. It rarely affects the enamel & doesn't cause cavities. Given in 1st & 2nd trimester, it can deposit in bones, growth of which is impacted only with long-term use, with "catch-up growth" after it's stopped. Studies since 1985 show no increased risk of congenital anomalies.
Intrinsic stain. If an expectant mom were prescribed the cycline family of antibiotics, then the babies permanent teeth may be intrinsically mottled when these teeth erupt. The color of tetracycline mottled teeth is not a surface stain. It is the inside and through & through color of the individual's teeth. These teeth may require aesthetic dentistry in the future.