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Rocket engines are a huge weight in a multi-stage rocket.
and they are stacked like so:
and so on, which means multiple engines.
Is there any issues using a staging system like so:
And route fuel pipes for the tanks down through the rocket.
There are some major challenges with this. For starters, the engines of the first stage produce far too much thrust for the last stage, which would require extra structural mass to allow the rocket to hold together, and would not allow manned rockets at all, as humans have fairly tight g-force limits. If you reduce thrust to manage these problems, you're carrying a lot of dead weight all the way into orbit, instead of just part way. So you need to, at the very least, stage most of the engine mass of the first stage.
There's also challenges in staging fuel tanks from between the engines and the payload: the tanks are structural! How do you get rid of part of the structure in the middle and allow it to reconnect safely and reliably?
Maybe you put the tanks on top, and just drop the top tanks in stages, but then you have more problems for manned rockets, as now the payload is trapped between engines and tanks, and astronauts can't use an escape tower. Plus you have to push empty tanks to one side, somehow, instead of just disconnecting them.
Maybe you put the engines above the tanks, but now you have to have higher-pressure (i.e., more massive) tanks to feed the pumps, and the engines have to be angled outward a few degrees or more for the exhaust to clear the tanks, which means you lose thrust.
Suppose you drop all this and just have parallel stages. This is the best approach, but still has some problems: frontal area is a lot larger than it needs to be, increasing drag substantially, plus the attachment points have to be stronger, and the tanks are shaped less optimally because they're too thin for their height. Plus, most of the rocket's flight time will be spent carrying multiple half-empty tanks, instead of a smaller number of completely full tanks and one partially-empty tank. So there's a number of mass penalties to this that make it less than amazing.
There's one final refinement to try: asparagus staging, also known as cross-feed. Falcon Heavy was originally intended to use a moderate version of this, but it was dropped due to lack of demand and engineering difficulty. Asparagus staging uses parallel stages, but pumps fuel from the ones that will be dropped soonest to the others, ensuring that tanks (and engines) are dropped as soon as the rocket as a whole uses up enough fuel, rather than waiting until each of the stages has used up enough fuel that the smallest are empty.
Unfortunately (you guessed it) this also has problems. Mostly the problems relate to the difficulties of getting cross-feed to work reliably and safely: the pumps have to be sized differently for each stage (even if the engines are the same), there's a lot of plumbing that has to reliably disconnect without damaging the exterior, and making sure the feed doesn't mess up the delicate pressure balance to the turbopump intakes is non-trivial. In principle these could probably be solved, with some modest mass penalties, but in practice it hasn't actually been worthwhile to do so yet.
Nathan's answer is good and cover almost everything but let me add a last bit:
An engine nozzle can be optimized for only one given altitude ambient pressure. This has a great impact on the rocket performance.
You probably wouldn't want to drag that heavy, inefficient engine all the way up, even if you could.
Some designs attempt to circumvent this issue like the aerospike engine, but are still prototypes.
Why not keep the engine for multi-stage rockets?
You don't want to carry useless engines all the way to orbit.
Rocket engines are a huge weight in a multi-stage rocket.
The premise is wrong. It's only the first stage engines that are heavy, the second and third stage engines are much smaller and of insignificant weight compared to the former.
You actually want to shed the first stage engines as soon as the rocket has become light enough for the second stage to take over. Just have a look at the Atlas rocket and its unique staging scheme. Atlas retained the fuel tanks but dropped two of three engines on the way up.
