You get 2Gb free which is pretty good and then use the public folder in your dropbox folder to store pictures you want to publish. Right click on a picture and then select "Dropbox/Copy Public URL" ..and stick this in-between the url tags.
The forum might consider using this image host. As it says: "The Simple Image Upload mod enables image uploading on your forum. All images are stored on our fast and secure network so it won't take up any of your bandwith." Find it here: http://postimage.org/mod.php