We can see from Plomp and Levelt's results that it is reasonable to see the presence of harmonics as something that has some bearing on how consonance and dissonance are perceived. However, in relation to your question, each of those graphs only represents a single data point ('no harmonics', and 'some harmonics') - which perhaps isn't enough to tell us everything about what the correlation would be.
As you've mentioned consonance, we want to pause to consider whether we want to consider 'consonance' as the absence of dissonance, or as a distinct concept. Does it make sense to consider silence, or a single sine wave, as 'consonant'? There's no dissonance there, but there's also no satisfying sense of consonance either. It may be that, rather than considering consonance and dissonance to be directions on a single axis, that we can consider them both as distinct phenomena that can both be observed subjectively in a given sonority - consonance being a result of the sine pairs that have simple (or near-simple) pitch ratios, and dissonance being a result of sine pairs that don't have those simple ratios (especially those within the critical bandwidth of each other).
We might therefore consider that the overall level of dissonance/ consonance is an aggregation of both the consonant and dissonant relationships in a sonority. Generally speaking, having more overtones provides more opportunities for consonance, but also more opportunities for dissonance. It might always be heard as 'richer', but the overall resultant consonance/dissonance is going to very much depend on the particular notes played, and the particular timbres in question - as well as the actual time/frequency behaviour of the overtones, as overtones in real life are rarely perfect integer multiples, and are often unsteady.
Is there a correlation between presence of harmonics and dissonance?
From the above, what I would guess would be that for a harmony that would conventionally be considered to be 'consonant', having more harmonic content would at first make the chord more cohesive and consonant, but then as the harmonic content increases further, it would start to get raspy, harsh, and dissonant. So there would be some 'optimal amount' of harmonic content for maximum perceived consonance.
If it varies case-by-case, please provide an example of when it does and does not increase dissonance.
Let's consider playing a closely voiced chord with a sinewave tone, and with a piano tone, in different registers. In both cases, the piano tone has louder overtones (as the sinewave has none).
If played in the octave above middle C, this chord is likely to sound more richly sonorous with the piano tone, because the ear can pick up on a greater number of consonant pitch relationships than with the sinewave tone.
If played three octaves lower, the sinewave version is likely to sound more concordant than the piano version. The lower notes of the piano have proportionately even louder higher harmonics, which have weaker relationships with each other on average.
