What resolution should my images on Tumblr be?

Depending on what Tumblr theme you're using, this will always be different. Always check your theme documentation to make sure your background images, logos, or post images should be a certain resolution or aspect ratio. But in general, these two rules usually apply:

Avatar images

Square avatars sizes are ideal, since Tumblr by default supports square images, for display in either square or circle containers.

Full-bleed background images

Background images are not usually dimension-specific but bigger is most often better. This means that your visitors will not be shown a pixellated or distorted version of your image on even the largest screens. Since most displays are rectangular, having a large image that's basically "desktop" shaped will be best, too. We would recommend somewhere in the region of 1300px wide by 800px tall, since most browsers run at that size these days.

Site logos

This will vary from theme to theme. Keep in mind that these will usually be smaller files, and you will want to either match the background color with your theme's background color or make the image's background transparent. Make sure that there is not a lot of excess whitespace around your logo, or it might be displayed too small within the logo space.