In a project of mine I have an event which was named "everything burns", or potentially "Everything burned" (I am open to both tenses). What would this be in classical Latin? I tentatively have this as either Omnia Urit or Omnia Exurit.
Your suggestions are not quite right, and they might in fact be badly misunderstood. There are two things to consider here.
The first one is simple. Omnia is plural and the verb must agree. Omnia (ex)urunt is grammatically valid.
The second and more complicated thing is ergativity. Some English verbs behave ergatively, meaning that the one experiencing the action is the object when both subject and object are present but the subject when there is no object. For example, in "I grow pumpkins" and "the pumpkins grow" the grower is always the pumpkin, but the vegetable seems to be the object in the first version and the subject in the second. One of the ergatively behaving verbs in English is "burn": In "I burn the wood" and "the wood burns" the same apparent switch of roles happens.
Other English verbs behave transitively. In "I kill the sheep" and "the sheep kills" the roles are not switched. It is always the grammatical subject that does the action and always the object (whether mentioned or not) that experiences it.
Latin verbs do not have this ergative behaviour. Instead, Latin verbs with an object are transitive. (There might be rare exceptions, but as a rule of thumb this is quite safe. And at the very least, awareness of the phenomenon will help avoid errors. See this earlier question about ergative verbs in Latin for details.) In particular, urere means "to burn [something]". That is, if everything burns (omnia urunt) using this verb, it means that everything is causing something else to be destroyed by flames. To express that everything is being burned, you need passive voice. What you need is omnia uruntur, with or without ex-.
If you want to switch it to the past, use the passive imperfect (if everything was burning) or the passive perfect (if everything has burned down). These would be omnia urebantur and omnia usta sunt, where sunt can be left out in some contexts. Without the sunt it can be read as "all burnt things" (a noun phrase) instead of "all things have burnt" (a clause); whether this is an issue should be judged on a case-by-case basis.
For a quick summary:
- uro = I destroy with fire
- uror = I am on fire
There are a number of verbs for burning, and they all behave this same way. In addition to urere, you can use exurere, accendere, incendere, and perhaps others I forget. (Thanks for reminding me of this, Rafael!)