Though I have no final reply to this, I try to elaborate a bit on this interesting, very good question. A similiar thread is here.
Assuming tx fees go through the roof, what happens with the funds, that are locked in a lightning channel? So let's go for an example: you open a channel for a micro payment (still lightning idea is to go with low values, I think there is a limit slightly above 1 mio Satoshi). The fees are low at this time, and you transact several times with your partner(s).
General idea of lightning would be, to NOT close the channel. Even if funds are exhausted (and you do not intend to reload), there is no necessity to close the channel...
Now you state, the channel shall be closed with an uncooperative node (that probably doesn't provide the service you payed for), and we assume, some remaining 10.000 Satoshis could be returned to me, though the fees for closing the channel are 100.000 Satoshis. Clearly this wouldn't make sense.
That is a dilemma, and cheating would be an activity for providers, that know, it doesn't make sense to close channel for the client, and thus "play" with this opportunity. It is becoming a reputation issue for the service provider then.
At this point I haven't seen any answer which tries to elaborate on it, when many service providers go down that road. As such it remains a risk, which is yet to be solved.
Could one leave the channel open and use it for different purposes?
One could think of stopping the payments (channel updates) to the uncooperative service provider, and use the funds for other services. Lightning is not bound as a 1:1 connectivity - it is a network, where e.g. your node can be used by others to hop to their target.
Maybe some lightning experts have a better view?