You'll need to use Photoshop or some other Pixel editing program. Lightroom is not a Pixel editing program.
Or you could crop the image to fit the paper.
It’s easy enough to set the paper size in the Print job tab – click image below – and you could then choose, say 12 x 8 to respect your 3:2 photo aspect ratio. It would be best to ask your print lab for a profile reflecting the color space and paper type they will be using. Otherwise ask if they can accept jpegs with a sRGB profile.
Use export plug-in LR/Mogrify 2 - Add Watermarks, Border and Text Annotions to Images Exported by Adobe Lightroom 2 powerful addition to rendering images of one aspect ratio into canvas of another. Once you are familiar with Mogrify you can batch process portrait and landscape photographs of different dimensions on to a 'standard canvas'. Many others variants e.g. borders captions backgrounds. Note if you are working with a PC you will need ImageMagick.
I want to be able to increase the canvas size for when submitting an image file for printing to a print service. The service requires that the submitted file exactly match the aspect of the paper to be used. For example, if I have an image file with aspect 3:2 and want to print to 11x14 paper
Option 1) At the risk of stating the obvious have you considered using the LR Crop Overlay tool with an 11 x 14 custom setting?
Option 2) Some print services offer 12" x 18" prints, which is the 3:2 aspect ratio of your original file.