Grace Morioka is usually dead on correct and complete in all her answers, which is why I like reading them. In this case however, there are some additional considerations that might make things easier than she suggests, so I'm starting with, Grace's answer is what happens most of the time, BUT, there are some exceptions that may get you around the problems she is pointing out. All of these are prefaced with "It depends":
First, it depends on whether you can do all plumbing changes within your walls or under your counters without changing anything within a wall that adjoins another unit. As Grace correctly points out, any changes made in a shared wall must be approved by the HOA board, which will require licensed contractor and possibly an architect to sign off on and will require, most likely a permit from the city. However, you usually own the space between the walls that are entirely within your unit. If the plumbing you need to change is within those walls, you MAY not need to even notify the HOA. This depends on what your Deed, CCRs and HOA agreements say, so read them carefully.
Second (actually third) it depends on whether you need to upgrade your electricity to 220 volt in a circuit or increase the load on a 110 circuit above its rated capacity in amps in order to accommodate the washer and dryer. If so, you need to go to the HOA and the city, just as Grace says. However, if you buy a washer and dryer that takes only 110 volts and you don't over load the circuit, then that is not an issue. In any case, if you are not electrically savvy, hire an electrician before you buy the washer and dryer to advise you. Overheated wires are the most common cause of home fires after smoking in bed and cooking fires in homes. If you cause the wires to overheat from such an installation, your insurance may not cover the cost of putting out the fire and the HOA insurance may insist on bankrupting you, as may your neighbors, especially if someone dies. In that case, the police and prosecutor may want to provide you with a very small room with lousy roommates and neighbors in a big house
Fourth, it depends on whether you can fit the washer and dryer into the unit without having to cut things up. There are many washer and dryer combo units that stack and some dry by centrifugal force instead of heat, eliminating the need for a flue or wet air exhaust with lint catcher and screen. There are tumbling (The barrel spins on a horizontal axis rather than agitating on a vertical axis to get the water and soap through the clothes) low water consumption combination units that are the size of a dishwasher that dry with centrifugal force. In Europe these are commonly installed under a counter next to the dishwasher. Any reputable appliance store should be able to tell you about current models. When installed under the kitchen counter next to the dishwasher, they use the same plumbing supplying the kitchen sink and dishwasher, resulting in no re-plumbing required. They can use the same electrical circuit so long as there is an A/B switch that does not allow both the dishwasher and clothes washer/dryer to run at the same time or if the circuit is upgraded. This depends on what your electrician advises.
Sixth, it depends on what the hazard insurance policy of the HOA and your own interior hazard insurance policy require, read both, let your lawyer advise you. Either or both may void coverage from damage or fire caused by the washer and dryer if the HOA is not asked before you install the washer and dryer or if you don't use a licensed contractor, or if you don't seek a city permit.
Even though your contractor, your electrician, your plumber, and your lawyer who advises you about what the CCR, HOA rules and insurance policies really mean, all agree you don't need to tell the HOA and don't need a permit, someone on the HOA board might get miffed if you don't seek their permission and don't seek a permit from the city. So it depends on how you want to handle that eventuality and how you want your relationship with the HOA to be.
So, as you can see, there are a eight "depends" you must consider to see if the exception to the rule applies in your case, which is why Grace's shorter answer is actually quite good.
Good luck, if I can be of any further assistance, say, in finding a home without this issue, I can be reached at firstname.lastname@example.org or by dialing 408-639-0211.