What does it mean if the controls in my IV model are correlated with my instrument?

This would not make instrument necessarily invalid. For some 2SLS instrument model of form:

$$y_i = beta_0 + beta_1 hat{x_i} + beta_2 k_i +epsilon_i$$

$$x_i = pi_0 + pi_1 z_i + pi_2 k_i +e_i $$

where $y$ is dependent variable, $x_i$ is the endogenous regressor, $k$ some controls and $z$ instrument the main conditions for instrument validity are:

• $z$ has to be correlated with the $x$ variable $cov(x,z)neq 0$ and have causal effect on $x$

• $z$ can affect the $y$ only through $x$ and $z$ itself has to be exogenous $cov(z,epsilon)=0$

Furthermore, instrument should not be weak which would invalidate inference from the regression and so 1st stage should have $F$-statistics higher then $10$ as a rule of thumb (although some new results suggest that actually correct rule of thumb would be having $F>105$ - see Lee, McCrary, Moreira, & Porter 2020).

You can find more rigorous proof of these conditions in econometric hanbooks such as Verbeek (2008) guide to modern econometrics or Angrist & Pischke (2009) Mostly Harmless Econometrics. The point is that none of the conditions directly say instrument cannot be correlated with other controls as long as it does not violates any other of the conditions. In fact omitting them could lead to omitted variable bias violating the condition that instrument should be endogenous if they are important. However, having highly correlated variables can inflate the standard errors of your model due to multicolinearity, this being said multicolinearity does not lead to bias it only reduces the precision of the estimates and reduces statistical power.

...the controls in my IV model are correlated with my instrument?

The controls should be in your model precisely because they are correlated with your instrument. In the exogeneity condition $cov(z, epsilon) = 0$ for the instrument, $epsilon$ is the error term after controlling for other variables. This condition may not hold without controls.

Consider the trivial case where $z = x$, i.e. the regressor $x$ is exogenous and is its own instrument. Then the controls are included in the regression precisely because they're correlated with $x$. Omitting the controls leads to omitted variable bias.

Considerations for inclusion of controls in the general IV setting is no different from the basic regression setting---e.g. issues such as omitted variable bias/multicolinearity(as mentioned by @1muflon1)/bad controls/etc.