The FU Orionis-type objects (FUors) are low-mass pre-main-sequence objects that go through a short-lived phase (~100 yr) of increased mass accretion rate (from 10-8 to 10-4 M ⊙ yr-1). These eruptive young stars are in the early stages of stellar evolution and thus still deeply embedded in a massive envelope that feeds material to the circumstellar disk that is then accreted onto the star. Some FUors drive molecular outflows, i.e., low-velocity wide-angle magnetohydrodynamical winds, that inject energy and momentum back to the surrounding envelopes and help clear the material surrounding the young star. Here we present a 12CO (3-2), 13CO (3-2), and 12CO (4-3) survey of 20 FUor-type eruptive young stars observed with APEX. We use our 13CO (3-2) observations to measure the masses of the envelopes surrounding each FUor and find an agreement with the FUor evolutionary trend found from the 10 μm silicate feature. We find outflows in 11 FUors, calculate their masses and other kinematic properties, and compare these with those of outflows found around quiescent young stellar objects gathered from the literature. This comparison indicates that outflows in FUors are more massive than outflows in quiescent sources, and that FUor outflows have a higher-ratio outflow mass with respect to the envelope than the quiescent sample, indicating that the eruptive young stars have lower star-forming efficiencies. Finally, we find that the outflow forces in FUors are similar to those of quiescent young stellar objects, indicating that their accretion histories are similar or that the FUor outflows have lower velocities.