I have heard that introducing capital gains taxes would create distortions in investments, and that the optimal tax rate for capital gains is actually 0. My question is what observed evidence is there of the distortions or "side-effects" that have been created do to the introduction of or the increase in capital gains taxes? Is there also empirical evidence of the opposite, i.e. an increase in investments (and any other "desirable" side-effects) due to the decrease or elimination of the capital gains tax?
Correcting Misconceptions in the Question:
Before providing an answer it is worth noting the premise in your question is simply incorrect. You state:
I have heard that introducing capital gains taxes would create distortions in investments, and that the optimal tax rate for capital gains is actually 0.
But this is not rationale for the famous 0 capital income tax Chamley-Judd result. Virtually all taxes create distortions, and any modern government, due the amount that modern governments spend, has to resort to distortionary taxation at some point. However, the famous Chamley-Judd result showed that (Chamley, 1986; Judd, 1985) the burden of capital income tax is in long run shifted to labor, so there is no point to those distortions and one should just levy those taxes on labor incomes instead (although more recent papers showed that the result does not always hold and that only some of the burden of capital income taxes gets shifted not all of it - e.g. see Straub & Werning 2014 or Diamond & Saez 2011).
So the 0 capital income tax result had nothing to do with distortions directly. Indeed taxing labor causes non-trivial amount of distortions as well and in spite of that those economists recommended taxing it directly.
Now to your main question on whether empirically capital taxation causes distortion, the answer is clearly yes. There are numerous studies showing that, and I think it is a fair to say that the question is not if there are distortions but how much and how economically significant they are.
The deadweight loss and distortions caused by capital income taxes will vary from country to country and from time period to time period, and it is beyond scope of a single answer to cover every country on a planet and every time period but here are examples of some studies that found capital taxes create distortions:
Devereux MP (2004) Measuring taxes on income from capital. In: Sørensen PB (ed) Measuring the tax burden on capital and labour. MIT Press, Cambridge
These are just few examples, but in public economics there is no question that capital taxes create some distortions (generally the only non-distortionary taxes are Pigouvian taxes and lump-sum taxes plus some special cases), so the question is not if but how much and answer to that will differ in time and place. Moreover, this has nothing to do with the zero capital tax result even pro-capital income tax economists would not deny taxing capital is distortionary.
For example, Goolsbee (1998) founds that corporate taxes (which are economically taxes primarily on capital incomes) in the US between years 1900–1939 caused distortions that resulted in deadweight loss of 5–10% of their revenue. Again that number will vary by place and time.
The positive effects of elimination of capital income tax on economy are elimination of the above mentioned distortions. Hence, if capital taxes caused in the US between 1900-1939 deadweight loss of 5-10% of revenue, then eliminating them would mean that this loss would not occur in the first place.