I am not sure if it is possible. But assuming it is, what would be somethings that this person would not be able to do?
I can't find any instance where a species or individual only had fast or slow muscle fibers, so I am unsure if it exists even with diseases. Something as simple as sitting or standing is an activity that is dominated by slow muscle fibers. Other activities like regaining balance, or moving quickly in any way are dominated by fast twitch muscles, so I don't believe it is possible to even survive on one or the other. Perhaps at one point some species had this mutation, but I can't find reference to one.
Fast VS Slow Muscle Fibers
Understanding what composes each type of muscle will be helpful, this previous answer does that well. The important part to take away from that answer is that fast twitch are ideal for short bursts of power where slow twitch are better for activities that require sustained use.
Can Fast Fibers do Everything Slow Fibers do?
Hypothetically fast twitch muscles can perform any activity that slow twitch muscles can, with the caveat that it can only do it for a short amount of time.
Perhaps you can jog on fast twitch muscles, but you wouldn't get very far before it starts getting painful.
You might be able to swim, but it is unlikely you could make it across an entire lake.
You can likely stand up, but you likely wouldn't be able to balance very long.