Year Fixed Effects in a Dynamic OLS Regression with Cointegrated Variables

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.

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

Inserting time (e.g. seasonal) dummies in also standard for stationary time series models. If the dummies are seasonal, such a component would be called additive seasonality.

Also, does it even make sense to control for year fixed effects when the independent and main dependent variable are non-stationary, but cointegrated?

It's not clear from your question the context that would motivate such a model but in principle one can consider such models.

(Presumably, you are considering the possibility that deviation from the long-run equilibrium relationship---the stationary error term in a cointegration regression---now has seasonality or time trend.)

In a cointegration regression $$ y_t = beta x_t + epsilon_t quad (*) $$ where $x$ and $y$ are I(1), and $epsilon$ is I(0), the OLS $hat{beta}$ is super-consistent, even when $epsilon_t$ is correlated with $x_t$. So if there is additive seasonality being omitted in the model and $epsilon_t = D_t + nu_t$, where $D_t$ is seasonal dummy and $nu_t$ is true error term, $hat{beta}$ remains super-consistent.

For testing for cointegration, however, specification may matter. For example, the Engel-Granger test computes ADF statistic of the residuals from $(*)$ (but with a different the asymptotic distribution under the null than ADF). If there is omitted seasonality, the asymptotic distribution can be different from that of Engel-Granger and their critical values may not be valid.