When you paste a link to an image in a DU post, the image is displayed directly, which is known as "hotlinking." In the case of PostImages, this means that the image is still located on PostImages' servers when displayed on DU, so whenever someone visits the DU page that contains the image link, it uses PostImages' bandwidth to display the image, not DU's bandwidth (because the image never resides on our servers).
Hotlinking is an incredibly common practice, but some free image hosting sites -- including possibly PostImages -- put a bandwidth limit on hotlinked images in order to reduce their costs. It's a little unclear as to whether PostImages does this, but the fact that the image initially worked, and was later replaced by the blue box telling you to upgrade to premium, suggests that they do.
This is why the problem is not related to your browser. It doesn't matter which browser you use to make your post -- the image will always appear initially with no problems, but once it crosses a certain bandwidth threshold (ie. after enough people have visited the page) then the image will switch to that generic blue box. Likely the only way around this is to use a different image hosting service that doesn't have a bandwidth limit.