Are the relativistic effects of gravity strictly due to spacetime 'curvature'? I happen to live on an infinite flat Earth.  It has a constant uniform gravitational field extending above it.  Don’t I still observe relativistic effects such as light not traveling in straight lines, even though there doesn’t seem to be any spacetime 'curvature' involved anywhere?
 A: This is a little bit of a semantic discussion: what qualifies as a “relativistic effect”.
Yes, light would follow a curved path, but light also could curve under Newtonian gravity. The relativistic deflection of light around the sun is twice the Newtonian prediction. In your uncurved spacetime they would be the same. So it is unclear to say if this is a relativistic effect or not.
There would also be time dilation. Time dilation is not explicitly part of Newtonian gravity, but Newtonian gravity is compatible with the equivalence principle. It is possible to derive gravitational time dilation from the equivalence principle. So it is also difficult to say if this is a relativistic effect.
I think a case could be made either way, but in the end whether or not you call them relativistic effects, the fact remains that there are many interesting gravitational effects in flat spacetime
A: 
I happen to live on an infinite flat Earth. It has a constant uniform gravitational field extending above it.

It's an unrealistic scenario, but I'm sure we've all heard of it. If the massive body is contrived as an "infinite plane", the strength of the gravitational field doesn't diminish with distance. See this mathspages article for some details: https://www.mathpages.com/home/kmath530/kmath530.htm

Don’t I still observe relativistic effects such as light not traveling in straight lines, even though there doesn’t seem to be any spacetime 'curvature' involved anywhere?

Yes you do. Light curves downwards wherever there's a gradient in gravitational potential. Or, if you prefer, wherever there's a "spacetime gradient". In the traditional rubber sheet depiction associated with Riemann curvature, light curves wherever the plot has a slope:

CCASA image by Mysid, see Wikipedia
Spacetime curvature is needed for a real gravitational field because without it, the plot above could never get off the flat and level. This is why spacetime curvature is a  "defining feature" of a real gravitational field. Unfortunately there's a bit of a myth that's grown up wherein people tend to say "light follows the curvature of spacetime", but that's not right, as your question demonstrates.
A: It is hard to imagine how a very big flat earth could not collape under its own gravity, becoming spherical. Maybe it is the case of say that spacetime curvature is a consequence of gravity instead of its cause.
But even on our spherical earth, GR tends to SR when spacetime interval tends to zero. It is the case for example of a radius of 4 km around (approximately the horizon line) and less than 1 km above us, for time intervals smaller that 1 minute. Everything passes in this case as we were is a ship upwards at an acceleration $g$. So all relativistic effects are there, but only SR equations are necessary.
