It can. A lot depends on how it is being constructed and the materials used.
A steel structure probable isn’t going to be affected; potentially a little rust but unlikely if the rain or snow doesn’t stick around for long and if it dries fairly quickly.
Wood is another story; there is depends if the wood is simply cut wood, or compressed, or kiln dried, and so on. The wood can absorb moisture and that can eventually waken it.
Is the drywall/plasterboard is up and gets wet, that is a bad thing, because it will weaken, fold, and potentially fall apart when wet.
Bottom line: it depends on:
1) material affected
2) length of time exposed to rain/snow
3) humidity