I don't mean like... voice messages, but, music! Specially blues, metal and rock which tend to have lower pitch tones.
The spectrum below 80Hz (generally called the sub-bass) will contain, at most, the thump kick drum and the fundamental frequencies/rumble of some bass notes, at least in the genres you mention. In some electronic genres, it may include most of the bass fundamentals (though not the harmonics -- it won't sound natural at all).
If you're talking about headphones that play everything above 80Hz, the inverse applies. You'll hear pretty much every part of the music (except, as before, some of the super low electronic bass drones you hear in certain genres of electronic music), but mostly it'll just be lacking punch and "weight" rather than any specific musical elements.
The question more likely is how much sound quality you will lose using generally bad headphones. For an ear-mounted speaker, reproduction of low frequencies is a lot easier to achieve than for free-standing speakers which have to fill a whole room (due to a lack of directivity at low frequencies).
So unless the headphones are intentionally designed to cut out low frequencies, you are likely to suffer independent degradation anyway.
Now there are monitoring speakers and headphones cutting off lower frequencies since they help comparatively little with staying in time or tune while putting additional load on the ears. Their purpose is not faithful musical reproduction but helpful musical reproduction. They are tools for the musician and sound engineer rather than the listener.