Will a budget amp damage my speakers or subs?

The bigger risk is clipping, not raw wattage. Underpowering a sub and turning the gain up causes a clipped signal that sends DC-like waveforms into the voice coil and burns it out. Reviewers across r/CarAV repeatedly point out that this is why a moderately overpowered, well-tuned amp is often safer than an underpowered one driven hard.