Given the polar nature of the N+--O- bond, one would expect low-polarity solvents to be considerably less effective--and actual NP solvents completely ineffective, as something like CCl4 would most likely be--as solvents than water or other H-bonding solvents. More important, considering that it is an N-*oxide*, it is an oxidant/oxygen donor (the latter being the predominant behavior, most likely), and I (and many, many, many others in the literature) have in used one (though quite dissimilar to DMT N-oxide) as a sacrificial donor catalyst for alkene epoxidation, alongside another particular oxidative catalyst. (In that particular case, the reaction is so fast that is has to be monitored using a controlled flow system. It had to be purified over a polar alumina column, which also suggests that limonene may not be the best choice as well.)
That is to say, it may react with the limonene and other olefin terpenes to form epoxides, similar to the behavior of certain peroxycarboxylic acids (which I have also used to that end--mCPBA is actually a typical reagent used to generate certain N-oxides!). However, the reaction in this case may be quite slow without another oxidative catalyst, depending on the pKa of the DMT N-oxide nitrogen, the temperature, the particular reaction kinetics, and several other factors, like whether or not (what appears to be) an unhindered allylic methyl moiety is more susceptible to attack. Be careful.