What would happen if archaebacteria would disappear

Everyone would die.

We'd immediately lose all photosynthesis and most of the energy production done by eukaryotes, since these processes are carried out by bacteria living inside eukaryotic cells (chloroplasts and mitochondria, respectively). This would be sufficient to knock out virtually all eukaryotic life. I suppose some archaea would survive and do well.

Now, maybe we should no longer count mitochondria and chloroplasts as bacteria, given that they've been living inside eukaryotic cells for several billion years. In that case, we would still be in deep trouble. We'd lose the nitrogen fixation abilities of Rhizobium, which would strongly affect agricultural production; various bacteria that help us and other animals digest food; most marine photosynthesis; and some other things.