I'm seeing conflicting opinions while googling about this. Is running with a training mask on beneficial in terms of improving VO2max? Can it be worn during comparatively light training activities like a tempo run?
tl; dr - Masks are (mostly) gimmicks.
They advertise as being able to raise VO2 max, by simulating high altitude training. This would in turn, raise EPO in the body, which then in turn stimulates red blood cell production, increasing oxygen transport.
In reality, while they may increase the muscle tone of the breathing muscles (intercostals, etc), they do not perform as advertised regarding increasing oxygen capacity. In fact, a couple of studies (admittedly with small sample sizes), have shown that maskless people training in the same manner/environment had better results than those wearing the masks.
Here are a couple studies you can read through, with the summary below:
Wearing the ETM while participating in a 6-week high-intensity cycle ergometer training program does not appear to act as a simulator of altitude, but more like a respiratory muscle training device.
This study concluded that the ETM did not cause a significant increase in VO2max under the training conditions of this study. However, results may differ if there is an increase in the frequency of exposure to the ETM, as well as an increase in the duration of the training period.
No-between group differences were found in any parameter (ANCOVA), therefore the two protocols should be considered equally effective.