Normally no.
You may just wish to read this:
http://www.epa.gov/dfe/pubs/comp-dic/lca-sum/ques4.pdf
where it shows that the major ozone contribution is in the power generation to manufacture and run crts (Cathode Ray Tubes) not in their use.
Now under what circumstances could you generate ozone.
Ozone can be generated by ionisation using high voltages. Well designed crts, although employing high voltages do not have any ionisation trails. The flyback transformer which generates the HV should have no naked HV points. There is normally a rubber cap where the HV cable connects to the tube.
Now if this cap is damaged or loose and there is dust accumulated around the connector you may have an ionisation trail and some ozone generation.
The possibility of ozone generation goes up with the HV required which is a function of monitor size. 15" monitors use much less HV than say a 23", if your pocket is deep enough.
So if you have a big monitor 17"+, and you're worried about ozone, check that the HV cap is well seated, and vacuum out the dust from the HV cap area and the HV transformer area occasionally.
Warning - do not poke around under the HV cap as there may be a significant charge stored in the CRT HV electrode. If the cap has to be replaced because the rubber is not well seated, this charge must be discharged before handling the connector.