Maybe a bit, though there are some very notable differences between the refresh rate of a monitor and what time dilation is like that you should be very cautious with.

First, refresh rate on a monitor reduces the number of discrete images the monitor displays per unit time. Time itself is continuous and not broken into fixed steps like that, as near as we can tell and we don't have any reason to think otherwise (note: you may have heard that Planck time is something like a discretization of time, but this is not the case. Planck time is a unit like seconds or decades and while some interesting physical phenomena are predicted to occur or have occurred over timescales similar to 1 Planck time it is not a discretization of time). With time dilation there's no stuttering or anything like that.

Second, the stuff displayed on the monitor is often flowing at the same rate, just with fewer frames. If I play a video game and it takes me 1 second for my character to run across the screen, that will occur in 120 frames at 120 Hz and 60 frames at 60 Hz, so that both take 1 second, but one is "smoother" as it has finer steps between each frame. Time dilation actually slows everything down relative to another reference frame. If it took my character 1 second to run across my screen from my perspective, it will take my character 10 seconds to run across my screen from the perspective of someone flying by me at 0.995c (even after kinematic redshifting is accounted for). Time dilation actually changes the speed at which time flows in one frame relative to another. It's not changing how long stuff takes in my reference frame, but how fast time itself is passing in my reference frame relative to a different reference frame.

Lastly, time always passes at the same rate in your reference frame: 1 second per second. Time dilation only make sense to talk about relative to another reference frame. If you're running by me at 0.995c, we will each calculate the other's watch as running ten times slower while seeing our own working fine, and that means we will disagree on whose watch is running slow. There is no absolute speed, so either of us can say we are stationary and it's the other one that's moving and be equally correct. Frame rate issues with a monitor are absolute as measured by the number of frames per unit time (in the monitor's reference frame, I suppose). Conversely, If I am very near a large mass and you are far from it we will both agree that my watch is running slow, but the fact that time is passing slower for me still only makes sense when compared to someone in a different location. In that case we know objectively who is closer to the mass so that gravitational time dilation is still relative, but is absolute in the sense that we can say for sure who is closer to the mass.

So yeah, kinda, but time dilation is different from lag that you'd have to be very careful in using that analogy. That said, all analogies are imperfect and the trick is knowing how and where they are applicable and where they aren't.