There are paints that are specifically for adhering to vinyl (i.e. what that jamb liner is made of). That said, the amount of friction that part gets exposed to, I am not sure that anything is going to stick to it long term and not look worn in patches after a few months of use.

Thanks to you both for giving me the proper term. I can't keep calling it "that plastic thing".

Windows on Washington, that's what I was afraid of. Whatever I try would just get worn off.

Toddinmn, I was thinking it was not safe to paint it either. Even though I really dislike the white. Switching to a tan jamb liner is not really an option: it would be way too costly and a lot of trouble. We had one set replaced when the windows were repaired. The contractor had a lot of trouble finding one that worked for these windows.

I'm not sure what you mean by "the edges of the window that slide against the jamb liner". That's the part you can't see, right, when the sashes are in their proper positions? The windows are designed to pivot and tilt out but I never do that.

Now I need to figure out how to remove the burgundy paint splashes. I can try Goof Off or acetone, and hope those don't damage vinyl.

I think you need to take a beat and reevaluate your choices here. It is perfectly acceptable to have lovely new stained oak trim around freshly painted white (I prefer a warmer white or off white) windows and doors. The paint color you have chosen is too close to "trying" to match what can not be matched. Paint does not, and will not look like stain--what you have done looks like covering up a mistake. The white jamb liner in between the painted window and stained trim only accentuates that. Paint your windows to match your jamb liner.

Also, please read up on DIY painting. The jamb liner should have been completely taped off, and all hardware removed, so that you did not have that mess of paint to try to clean up now. Be very careful at this point that in your attempts to remove paint, you don't also muck up that new oak finish. It is totally possible to clean this up, turn it around, and make it look great! Just not with that paint color, and not without significant prep work before you even get the brush out again.

Yes , that what I mean. When people repaint the sashes often the get it all over the edges. This can make them difficult to operate.