I read that it is a good idea to use mirror lock up to avoid camera shake.
What are to disadvantages of using mirror lock up?
I enjoy street photography, using a Nikon 28-70 mm lens and where possible use a wired remote.
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The only disadvantages are that it slows down your shooting (you have to wait a second or two after locking the mirror for the vibrations to subside) and that the viewfinder is black during this time so you can't react to changes in the scene.
Having said that, you only benefit from mirror lookup when using telephoto lenses, or when shooting slow shutter speeds (around a quarter of a second). In these situations you'd be shooting with a tripod so the slow down is tolerable.
If you want to get steadier shots with a 28-70 lens then I would advise adopting a wide stance, locking your elbows, bracing the camera against your face and shooting in bursts.
Mirror lock up is effective if the camera is held in place (by a tripod, resting on a place...). If the camera is hand held, or on a monopod, its benefits are reduced or completely lost.
In short, when using mirror lock up, first you compose, then the mirror flips and the camera waits for some short time before taking the picture. This gives time to dampen vibrations, which is useful for longish exposures (up to about a couple of seconds). Note that for longer exposures vibrations occurring only in the first seconds are much less relevant.
Mirror lock up can be very useful for astrophotography (where you tend to have longer exposures and a mount). For street photography I think that with the help of a filter to allow for longer times you could find it useful: since you mention a remote this could be the case.