Does DPI matter for web images?

Not in a meaningful way. Web browsers render images by their pixel dimensions and ignore the embedded DPI tag entirely, so a 1200 x 800 image looks identical whether it is labeled 72 or 300 DPI.

• For screens, focus on: pixel dimensions (actual width and height) and file compression for fast loading.
• DPI only matters for print, where it maps pixels to a physical size on paper.
• Watch file size: a needlessly large image slows page loads without improving how it looks on screen.

Save the 300 DPI thinking for anything headed to a printer, and size web images to the largest dimensions they will actually display at.