Tropospheric (near the surface) ozone is a powerful greenhouse gas, even in trace amounts. Strataspheric ozone or the ozone layer is opaque to UV rays coming in and it's opaque to IR rays going out so it has both warming and cooling effects. The net effect of a thickening of the ozone layer is a small warming (with some uncertainty), so the ozone layer is not a very strong heat trapping driver, but ozone in the lower atmosphere is, and it's a strong heat trapping gas one too.
Ozone is very reactive and as a result, it has a very short atmospheric lifetime. (hours-days), but what Ozone has is an equilibrium concentration in the atmosphere (337 ppb). It's concentration is maintained in the troposphere by sunlight hitting trace elements in the atmosphere.
The majority of tropospheric ozone formation occurs when nitrogen
oxides (NOx), carbon monoxide (CO) and volatile organic compounds
(VOCs), such as xylene, react in the atmosphere in the presence of
sunlight.
NOx and VOCs are called ozone precursors.
Motor vehicle exhaust, industrial emissions, and chemical solvents are
the major anthropogenic sources of these chemicals
If, for example, you were to release several billion tons of Ozone on the surface of a planet to increase it's greenhouse effect, the effect wouldn't last long because because ozone is for the most part an unstable molecule. It needs to be constantly recreated because it tends to chemically react and stop being Ozone very quickly for an atmospheric gas. As noted in the quote above, motor vehicle exhaust has increased the atmospheric material that ozone gets created from in our lower atmosphere, so there are man-made reasons for the increase in tropospheric ozone concentration, 237 ppb in 1750 to 337 ppb today.
This reference from the Wikipedia article, notes that the Ozone layer has a minor cooling effect, (Source article here), quote from footnote 6
Radiative forcing for tropospheric ozone is taken from the 5th column
of Table 8.6 of IPCC (2013).
http://www.climatechange2013.org/images/report/WG1AR5_Chapter08_FINAL.pdf
The "current" value in that table refers to a global average. Note, in
the row immediately below the number for tropospheric forcing, the
stratospheric forcing is given as negative 0.05 W/m2
The 82 page IPCC report on radiative forcing from which that article references has this to say on it's introductory remarks on Ozone:
The total RF estimated from modelled ozone changes is 0.35 (0.15 to
0.55) W m–2, with RF due to tropospheric ozone changes of
0.40 (0.20 to 0.60) W m–2 and due to stratospheric ozone changes of –0.05 (–0.15 to +0.05) W m–2. Ozone is not emitted directly into the
atmosphere but is formed by photochemical reactions. Tropospheric
ozone RF is largely attributed to anthropogenic emissions of methane
(CH4), nitrogen oxides (NOx), carbon monoxide (CO) and non-methane
volatile organic compounds (NMVOCs), while stratospheric ozone RF
results primarily from ozone depletion by halocarbons. Estimates are
also provided attributing RF to emitted compounds. Ozone-depleting
substances (ODS) cause ozone RF of –0.15 (–0.30 to 0.0) W m–2, some of
which is in the troposphere. Tropospheric ozone precursors cause ozone
RF of 0.50 (0.30 to 0.70) W m–2, some of which is in the stratosphere;
this value is larger than that in AR4. There is robust evidence that
tropospheric ozone also has a detrimental impact on vegetation
physiology, and therefore on its CO2 uptake, but there is a low
confidence on quantitative estimates of the RF owing to this indirect
effect. RF for stratospheric water vapour produced by CH4 oxidation is
0.07 (0.02 to 0.12) W m–2. The RF best estimates for ozone and stratospheric water vapour are either identical or consistent with the
range in AR4. {8.2, 8.3.3, Figure 8.7}
From page 3 of this link, chapter 8, page 661 of the AR5 report.
What the report basically says is that tropospheric ozone has a significant greenhouse effect, 0.4 watts per square meter as a result of the man-made increase since 1750, which is nearly equal to the warming from the methane increase, despite measurably lower total concentration and a lower percentage increase. (note, there are considerable measurement uncertainties, 0.2 to 0.6 is given as the range). But even the low end of that estimate makes ozone in the lower atmosphere a powerful greenhouse gas.
The Ozone layer (0.05 watts per square meter cooling estimate) corresponds to a thinner ozone layer today compared to 1750 (as far as I can tell, I didn't see that spelled out), but assuming that's the case, a thicker ozone layer should (could) provide a small amount of warming, but 0.05 watts per square meter variation is pretty insignificant.
