Hilchos Shemittah v’Yoveil 5:17

It is permitted to chop down trees for wood during shemittah before they produce fruit. Once they have started producing fruit, they may not be cut down. This is because doing so would waste the fruit, and shemittah produce was given us “for food” (Lev. 25:6), not to waste. If a tree produced fruit that has reached the tithing stage, then it may be cut down. Since it has already produced its fruit, the laws of shemittah no longer apply to it.

Hilchos Shemittah v’Yoveil 5:18

The prohibition against chopping down a tree during shemittah applies as follows: for carob trees, when a chain of carobs forms; for grape vines, when the grapes grow seeds; for olive trees, when they bud. All other fruit trees become prohibited once they produce undeveloped fruit. One may not cut off undeveloped date clusters during shemittah because doing so wastes the fruit. If these clusters won’t produce dates, then they may be cut off.