I have seen a number of places that seem to say that hydroponics is mutar during shmittah. If this is the case for plants that normally grow in the ground (such as tomatoes), then the algae in your fish tank would also be mutar. The same question would apply to a fish pond.
Note in the footnote below that there are those who say it is forbidden. Also I am assuming from your question that you are in Eretz Yisrael. Otherwise, the question would not apply.
Analyzing the Roots: Hydroponics and Halacha
Due to the special status of Eretz Yisroel, generally,
vegetables grown there must have ma’asros, tithes, separated.
However, if the AeroGarden kit is used to grow vegetables in Eretz
Yisroel, ma’asros need not be separated from the vegetables. This is
because hydroponically-grown vegetables are free from the obligation
to separate ma’asros.11 In addition, they may be grown during the
Sabbatical year of Shmitta.12
On a related note, if the seeds in the kits were to come from Eretz
Yisroel, the seeds would not be subject to the laws of ma’asros or
Shmitta. These seeds are not edible seeds, and the Mishnah in
Ma’asros 5:8 states that non-edible seeds are not subject to ma’asros
or Shmitta.13 In any event, Star-K has determined that the AeroGarden
seeds are not from Eretz Yisroel.
- Yechaveh Da'as ( 6:12) citing Rav Yechiel Michel Tuchetzinski (Sefer Ha’Shmitta vol 2:pg 104), Sefer Nechpeh B’kesef (Y.D. 1:5)
cited by Minchas Shlomo (vol 3, 158:8), Rav Binyamin Zilber (Az
Nidbiru 4:51). Rav Zilber emphasizes that his ruling applies only in
cases where the plant is not in any earth or even chemical
fertilizer. The Chazon Ish is also quoted as having permitted
hydroponics during Shmitta. Maharil Diskin, cited by Minchas Shlomo,
forbids all hydroponics during Shmitta, as does Har Tzvi (Zeraim