How do you install shiplap over plaster walls?

If it is true plaster, there should be boards behind it with only a small crack between then chicken wire (or similar) and then plastered on top of that. If the plaster isn't terribly thick you can attach through the plaster into the boards behind. You might want to do a bit of chipping away to investigate which is behind it.

You will need to anchor them to the studs. I would use some construction adhesive on the shiplap and nails to anchor to studs.

Hi Tami, here are some hints on installing ship lap over plaster walls. Good luck with your project.Can you install shiplap over plaster walls?

When installing shiplap over plaster or hardboard, the walls need to be in moderately good condition. Construction adhesive is usually the easiest and fastest way to install wood shiplap over drywall, plaster, hardboard and even old shiplap. When applying adhesive, be sure to follow the manufacturers instructions.

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rustic shiplap installation instructions - UFP-Edge

You can do a faux shiplap with a lighter material. Then it can be held on with adhesive and nails. You Wont need to worry about studs etc. We did something similar: https://www.frugalfamilytimes.com/2018/05/how-to-make-vertical-shiplap-paneling.html?m=1

Plaster can be tricky...perhaps this will help. https://www.stretcher.com/stories/03/03jul28h.cfm

Hi Tami, hope this helps... https://www.hometalk.com/diy/decorate/rooms/q-how-do-i-shiplap-over-plaster-walls-33358126

When installing shiplap over plaster or hardboard, the walls need to be in moderately good condition. Construction adhesive is usually the easiest and fastest way to install wood shiplap over drywall, plaster, hardboard and even old shiplap. When applying adhesive, be sure to follow the manufacturers instructions. To make sure you have a tight joint when applying adhesive, use small nails to tack up and hold the shiplap in place on the drywall or plaster while adhesive sets. Apply tack nails every two or three feet. DO NOT NAIL THROUGH OVERLAPPING PIECES

You should figure out where the wall studs are by using a stud finder. Once that is done then you can apply your shiplap with screws or long brad nails. Shiplap can be heavy so needs something more solid than drywall to nail to,

I think i would put up two trim boards on each side of the space between the wall and fireplace, and adhere the faux shiplap to them so not to tear up the whole wall. It will be much easier to fill the holes when you tire of it. The trim boards are usually 1-2" wide.