Will the 595 shift register burn out if outputs are always high?
There should be almost zero current being drawn from the 595 while the outputs are high, so it shouldn't be an issue.  If the relays were active on a HIGH, then you would have to look at the current being drawn from the 595 for the relays coils and, using the datasheet specs, compare that to the "per output" MAX current compared to the current that the relays required.   There should be per pin and per package current spec. If the total amount of current being sourced or sunk does not exceed the 595's datasheet specs, you will be fine.