If its dying it wont transplant well. Just cut it down and apply boiling water to the roots. later, you can pull the roots out.

I just dug out some old shrubs... must have been 50 years old. Azaleas came out easily but the evergreen shrubs (not boxwoods) were awful. We tried to pull one out with the truck but snapped the nylon tow strap. We just cut the rest down & have been digging out the roots one by one. Four down, one to go! Dug out all around and ended up sawing some of the roots off. It would have been easier to pull them out so you could try it that way.

Depends a lot on what kind of bush it is -- some come out easily and some not so much. But agree with the general strategy: cut off branches, water well (or wait for a nice rain storm!), and try to dig out roots.
Azaleas: easy
Holly: awful...

I have the same question, OP as I am wanting to remove all of the ugly bushes in the front of my house and replace them but am concerned we won't be able to get the entire bushes and their roots out.

A co-worker rented a pick up truck from Home Depot and attached a chain to the truck and around the bushes and pulled them out that way!

OP here. Thanks for your advice, PP.

I have no idea what the shrub is. I think maybe a big boxwood. I think it's dying because we never watered it. I thought it would get enough from the rain alone, but probably not; that side of the front of the house gets quite a bit of sun. I'll definitely ask for something low maintenance when we re-plant. And water it.

Are there tools that will help with root removal? (Beyond normal shovels, which we have, but something we could rent or that isn't too expensive.)

You can do this, but wait until fall. You don't want to plant a shrub in hot weather.

Getting the shrub and all its roots out will be the tough part.

Cut back as much as you can now to make it more manageable.

Also, if it is very dry or compacted, water the area regularly to help removal.

Replanitng is the easy part, getting it out takes muscle.

What shrub is this, btw and why is it dying? This will help you in choosing your next plant.

A very large shrub in front of our house is dying and needs to be removed. It's about 3'x10'. DH and I are both very novice gardeners, but I don't want to spend the money to pay someone else to do it (the front yard needs an overhaul that's not in the budget right now, but this shrub looks awful and I'd really like to deal with it immediately; however, I'm not sure that whatever we do now will work with the overhaul that we do later, so I'd rather not sink money into a project that I'm just going to undo in two years).

Is this something that novice garderners could handle? Not just removing it, but also replanting something in it's place? Or are we just going to screw it up and we should save ourselves the effort and hire someone to do it right?