Dangers of Colbalt Chloride Paper Burning
Not a good story to share ...


It is not surprising that a rapid heating of a Petri dish provided by a Bunsen burner cracks the former.  By application of the heat, the glass wants to expand, to dilate; hence you swirl test tubes over a Bunsen, provided your lab still has access to natural gas.
A more gentle heating of the glass, to dry the paper impregnated by $\ce{CoCl2}$ were gradual heating on a heating plate.  By this you equally would obtain the colourless, waterfree form, in a more safe manner (no open flames, no ignition).
Yes, $\ce{CoCl2}$ is considered toxic, and dangerous for the environment.  It is suspected carcinogen. You set yourself unnecessarily in danger, not only because of the flames, but that the dry salt, too: finely dispersed in air, it is then easily inhaled.  You won't die on the spot, however, do not repeat this. And ChemSE is not a subsitute for a physician. 
Depending where you did this, disposal of heavy metal, and aqueous solutions of heavy metals (like $\ce{CoCl2}$) is regulated by law, and hence the rinse with water was probably fine to extinguish the fire, but the waste water generated is not normal household waste.  Especially if it were on a larger scale.
For future reference: look up safety informations of chemicals you intend to work with, chemical suppliers include them in MSDS sheets. In chemistry lab classes, students are requested to get familiar with these information prior to performing the experiments.