We Just Got A Blue Heeler Pup And His Eyes Are Running. Is It An Infection Or Allergies?

Unfortunately this is one of those problems that has no easy answer... The eye's natural responses to irritation is to produce more tears... This helps flush away any irritants that might have contaminated the eye surface.  Allergies, irritating dust or smog in the air, injury, or infection can lead to excessive tearing... If there's squinting or pawing at the eye, or if the  tearing occured suddenly (did he have the condition when you got him?), this is a sign of eye pain and you need to see a vet as soon as possible. 

The more chronic cases are of course harder to manage... Eyes can, and do, become chronically irritated from viral conjunctivitis.. From glaucoma, or more commonly from eye or eyelash abnormalities
Hairs can grow from the face onto the eye and cause irritation... Eyelashes can grow at abnormal angles and rub on the eye, or can even grow on the inner surface of the eyelids and cause irritation... Magnifying instruments are needed to find these tiny hairs and minor surgery is needed to deal with them.
Once you've ruled out a painful condition... then chances are the drainage ducts of the eyes may not be normal or working properly.  There are many problems that can occur along this drainage route... The normal eye is very efficient at draining tears... In the inner corners of each eye is a tiny hole that leads to tear ducts.. They drain the tears into the nasal passages and throat. If these aren't functioning properly, the tears can't drain and so they spill down from the corners of the eyes.  Another common problem for some breeds (not normally healers) is that the eye socket is shallow. This means that tears over flow from the corner of the eye because the eyelid space there is just not deep enough to contain them....if this is the problem, it can't be fixed... it's just the shape of the dog's face.