Events that are not mutually exclusive can be dependent.

Example: drawing a King or a Heart from a deck of cards. This is not a mutually exclusive event: if you draw a King, that doesn't rule out the fact that you haven't drawn a Heart. You might've drawn a King of Hearts.

$P(K\vee ♥)=P(K)+P(♥)-P(K♥)=4/52+13/52-1/52=4/13$.

You draw a card and see that it is a King but don't see the suit yet. Even though you don't know whether the event ♥ happened, because the event K happened, the probability has changed to $P(K\vee ♥)=1$ (Technically, $P(K\vee ♥|K)=1$, the probability of "K or ♥ given K").

If knowledge of event A changes the probability of event B, the two events are dependent.

The knowledge that you drew a King changes the probability of the event $K\vee ♥$ (drawing a King or a Heart), therefore the two events are dependent.

(If you drew a card and saw that it was not a King, the probability would have changed to $P(K\vee ♥|\neg K)=3/13$; regardless of the outcome, this is still a dependent event.)