Randy Subers said:
Although the NASA page you link to claims this work is based on peer-reviewed science, there are no links to actual peer-reviewed papers in it. So there is no real basis for discussion there.
The Wikipedia page for James Woodward does have some links to papers:
https://en.wikipedia.org/wiki/James_F._Woodward
The first one (ref. 2 in the article, the 1990 paper) appears to be claiming that the effect on which the MEGA drive is based is either the same as, or similar to, the Nordtvedt effect. Unfortunately, there are very tight experimental limits on that effect, as shown in the references in this article:
https://en.wikipedia.org/wiki/Nordtvedt_effect
I have only skimmed the papers so far, but my general sense is that claims are made about GR which are based on misconceptions and are not actually correct theoretically.
That said, it is perfectly possible that experimental effects exist for which we don't have a good theoretical understanding; that has happened many times in the history of physics. However, looking through the long PDF linked from the NASA article (which is not a peer-reviewed paper, it's a report to NASA from the grant recipient), it seems to me that the effects observed so far are tiny and quite possibly due to experimental error. A real demonstration would have to involve a clear production of an easily measurable force under conditions where there is no possible source of that force based on present-day understanding (no rocket, no laser beam, no wind, etc.). It's not clear if or when NASA will be trying to make such a demonstration for this proposal.
Randy Subers said:
It seems to violate conservation of momentum.
Momentum can be stored in fields, and the basic hypothesis on which this proposal is based appears to be that there is some kind of scalar field associated with gravity (the general term for such theories of gravity is "scalar-tensor theories"), which could serve as a source of momentum (just as electromagnetic fields can store momentum and impart it to charged particles). That's not to say that that hypothesis is true, just that it is not necessarily open to such a simple objection.