I have RNA-seq data from a 3-level 1-factorial experiment: non-treated control, placebo-treated negative control, and treated cells. I have 3 replicates for each.

After running edgeR, and also looking at expression levels, I noticed that the negative control is not a good control, as it has DE genes compared to non-treated, while the treatment is not DE for these genes (has the same expression levels).

So to be safe, I want to find genes that are DE compared to non-treated and also for negative control.

How should the contrast matrix be designed?

I suspect that the solution is to simply take the intersect of separately calculated DE gene lists (instead of building it into the contrast).

My design matrix:
groups <- factor(c(0,0,0,1,1,1,2,2,2))
design <- model.matrix(~ 0 + groups)
colnames(design) <- c("O", "N", "T")

You are correct. The solution is to perform the individual contrasts separately and then take the intersection.

I agree with Ryan's support of the 'intersection' approach you suggested, in cases where you wish your effect to be distinguishable from both types of control, but you'll be subject to two rounds of exposure to statistical errors, with the subsequent loss of power.  I think it might be worth looking at other approaches also, depending on the definition of 'placebo' here.  If the placebo is inducing some partially confounding effect (a scrambled siRNA vector,...) then that is the true control, as your three conditions are then O=baseline, N=baseline+placebo_effect, T=baseline+placebo_effect+biological effect, and N-T gives you the straight biological effect, and you can effectively ignore the untreated group (apart from it's contribution to the estimation of within-group noise).  Eliminated things that don't have an O vs T significant effect could in this situation decreases your power and potentially introduces bias against genes where the biological effect is working to counteract the placebo effect (only the experimenter will know if a gene that is upregulated in response to placebo, and reverts to baseline on full treatment is interesting or not).

At the other end of the spectrum, you could actually pool your two types of control groups <- factor(c(0,0,0,0,0,0,2,2,2)) so that you'd be including any placebo effect as part of the 'replicate' variability, and anything that survives this increase in variability is a sufficiently large biological effect that it dwarfs any placebo-induced variability.

When I get vaguely-specified controls, I tend to apply all three approaches, and draw the expression profiles of genes that are significant in some but not all of the approaches - the experimenter then has a clear visualisation of the different hypotheses being tested.