Year Fixed Effects in a Dynamic OLS Regression with Cointegrated Variables I am estimating a dynamic OLS model since I have variables that are non-stationary, but cointegrated. In addition, the data is a standard time-series (i.e. one observation per one time period) so technically not panel. I know that one should control for year fixed effects when you have panel data. My question is should I still control for year fixed effects even though my data technically isn't panel. Also, does it even make sense to control for year fixed effects when the independent and main dependent variable are non-stationary, but cointegrated?
No in pure time series we generally don't use fixed effects. If you have data on lets say monthly frequency you could include dummies for months in general, e.g. having February, March, April ... dummies but you would include them as a general dummies where all Februaries are part of the February dummy rather having a separate dummy for each time period.
You can also include dummies for time periods where you think that the period was special in some way. For example, you could include a special dummy for all crisis years. Similarly, you could include some dummies for structural breaks and so on.
Furthermore, dynamic OLS model is more of an umbrella term. For example, the above would hold for ARDL/Pesaran error correction model. There might be dynamic models that have further issues, it is always best to survey literature on the particular model you are actually using then just generalizing.
PS:

I know that one should control for year fixed effects when you have panel data.

This is actually not generally valid statement. Yes more often than not you want to control for year  fixed effects in panel data but not always.