MySQL - why not index every field?

You have to balance CRUD needs. Writing to tables becomes slow. As for where to draw the line, that depends on how the data is being acessed (sorting filtering, etc.).

Indexes take up space in memory (RAM); Too many or too large of indexes and the DB is going to have to be swapping them to and from the disk. They also increase insert and delete time (each index must be updated for every piece of data inserted/deleted/updated).

You don't have infinite memory. Making it so all indexes fit in RAM = good.

You don't have infinite time. Indexing only the columns you need indexed minimizes the insert/delete/update performance hit.

Keep in mind that every index must be updated any time a row is updated, inserted, or deleted. So the more indexes you have, the slower performance you'll have for write operations.

Also, every index takes up further disk space and memory space (when called), so it could potentially slow read operations as well (for large tables). Check this out