Question: Is it permissible to still eat in the Sukkah on Simchat Torah?
Answer: What a fantastic question.
The Torah commands us to sit in the Succah for Succos, not the day after Sukkot which is Shemini Atzeres. And there is a prohibition against adding onto the Mitzvos of the Torah. Thus, the Talmud asks how we can sit in the Sukkah on Shemini Atzeres. The response is that we can sit in the Succah on Simchat Torah, but we should not make a blessing. The Talmud proceeds to explain that this is not actively adding onto the Mitzvah as people often sit on their porches, therefore they are not clearly sitting there for a mitzvah. They are sitting there because they enjoy it.
Those of us who live outside of Israel keep two days of Yom Tov due to questions about whether the new moon was declared. Since we do not know when the holiday really should take place, the Rabbis decreed we should keep both possibilities. So since we are unsure if Sukkos is over and Shemini Atzeres-Simchas Torah has started we need to make sure we are not adding onto the Mitzvos.
So, coming full circle, can we continue to sit in the Sukkah on Simchat Torah?
This topic has been very extensively discussed because virtually all Jews of Chasidic descent do not sit in the Succah on Shemini Atzeres!
As we explained, spending time in the Sukkah on Simchat Torah is not necessarily adding to the Torah Mitzah to dwell in the Sukkah during the Sukkot festival. I do not recommend sitting in the Sukkah on Simchat Torah because it is difficult to differentiate between "sitting" in your Sukkah (adding to a Torah Mitzvah) from just enjoying the "porch". Your question reflects how much you enjoy the Sukkah; not just the "porch" aspect.
The other issue associated with eating in a Sukkah on Simchas Torah is a custom quoted in Shulchan Aruch (the code of Jewish Law) that we do not eat in planted areas on Shabbos or Yom Tov to avoid watering the ground with spills.
For all these reasons I would not recommend eating in your Sukkah on Simchas Torah.
Wishing you a happy Succos,
Rabbi A. Page
P.S. There are some fantastic discussions of the holiday at Torah.org and a very well done book at ArtScroll.com. Artscroll has a series of exceptional works on the holidays, but I happen to like this one in particular. If you can get ahold of it, some of the best essays I've ever seen on the holidays are in Seasons of the Soul. Another extremely enjoyable book on the Jewish holidays is A Gift For Yom Tov.