Is Jet Fuel/Kerosene still toxic after drying out?
Kerosene is not a pure compound. To quote Wikipedia:

Regardless of crude oil source or processing history, kerosene's major components are branched and straight chain alkanes and naphthenes (cycloalkanes), which normally account for at least $70~\%$ by volume. Aromatic hydrocarbons in this boiling range, such as alkylbenzenes (single ring) and alkylnaphthalenes (double ring), do not normally exceed $25~\%$ by volume of kerosene streams. Olefins are usually not present at more than $5~\%$ by volume.[10]

[10]: American Institute of Petroleum (September 2010). &quot;Kerosene/Jet Fuel Assessment Document&quot; (PDF). EPA. p. 8. Retrieved October 28, 2016.

Because kerosene is a complex mixture, it is hard to say something specifically. However, the compounds noted are basically long chain alkanes and substituted cycloalkanes, a smaller fraction being alkylated aromatic rings. Benzene and naphthalene do not seem to be significant contributors.
None of these are classified as toxic. Most importantly, while alkylated benzenes and alkylated naphthalenes may still intercalate into DNA making them potential carcinogens, they are metabolised by a completely different pathway from benzene proper, making their metabolites much less dangerous. (Rather than the very reactive 1,2-epoxybenzene, various benzoic acids are the primary oxidation products.) That doesn’t mean they’re harmless, but toxic is the wrong designation and should be reserved for more dangerous compounds.

A general rule for all compounds is:

As long as you smell them, they can affect you.

The desired effect of painting a door with kerosene is, however, that the kerosene penetrates the door and stays there. The least-volatile constituents will remain on the door itself, protecting it from damage. If all the kerosene diffused away, the method would not be used because it would be worthless. Therefore, after the door has vented properly, letting the volatile components disappear, there is practically nothing left that will be harmful to any noteworthy extent.
In fact, I am include to proclaim that your average coloured paint may well contain more harmful chemicals that the kerosene does.
Yes, you can touch it, no you do not need space suits to be safe.