Shiplap never , do you really know what shiplap is most think it is just smooth boards and that is not the case.BTW I would be removing the carpet and replacing before thinking about the fireplace.

Agree: " Shiplap never" -- especially adjacent to a fireplace or other direct heat source or serving as paneling without sheet rock behind it. Boards subjected to heat shrink and leave spaces between them … which is why better quality paneling and flooring has the tongue & groove edge.

As a safety issue, do remove all carpeting in a room with a fireplace if wood is ever going to be burned it. If, however, that's a gas fireplace, sparks wouldn't be as much of a problem -- but tile or plank flooring would probably be my preference anyway.

I'd also advise against shiplap (low-quality boards, the grooves get dusty and it's been a popular trend that's now fading), but if you love the look, you can do it around a fireplace provided you have enough separation with your surround (local codes usually specify this) and make sure you have a good installer who uses good quality materials. In terms of what comes first, it depends on how you are going to handle the transitions between hearth/surround/wall. I'm guessing tile in your case. A sketch or picture of how you envision your final product may be helpful.

You tile the whole thing first. Then you take the shiplap back to Lowes.

Wow! Okay. Back to the drawing board. We weren’t going to do actual shiplap but the tongue and groove.

For what it's worth, my inclination would be to put tile first all the way to the floor plate to cover/protect the sub flooring, then add any sheetrock or paneling to the wall. If there is a small space at the bottom of the wall covering above the tile, that could be covered by your baseboard.

It's going to be funny when all the HGTV shows in a few years show homeowners tearing out all the shiplap that got installed in the 2010s... What a terrible trend, there's a reason everybody had stopped using it in the first place.