I've heard that at some beaches in Zanzibar the sea completely "disappears" at low tide and one can spend half the day without any sea. Is it true that Northern beaches have the least tide difference?

Believe it's north and west.
But even on the east, as tides are moon dependent, you could luck out to have a few days with little tidal effect.

as a general rule the whole island is very tidal, but there are some locations - such as Pongwe – that, because they are in a cove, are much less affected by the tides. This is less to do with the side of the island and more to do with their specific beach aspect.
Also if you take somewhere like Nungwi village and Ras Nungwi hotel (NW and NE) of the island and only about 10 mins drive from each other, if the tide is out at one it is in at the other ...

So should we be checking tide tables to plan our trip? I've done this for other locations. If so, any suggestions on a link to the correct table?

Don't know about you, Snookie, but I'm now thinking for any Zanzibar trip I must coordinate the timing of elections, dolphin pod movements, water clarity for snorkeling, the whale shark migrations, the flying foxes that are really bats, visibility of various monkeys, good safari conditions in the rest of Tanzania, minimal rain, and temperatures that are not too hot but warm enough so I don't shiver while participating in the water activities, ALONG WITH the tide tables. After all that, I may need to just lay on the beach!