Hetzner's ToS and above cited comment on Reddit are a bit vague and I read it that they disallow anything related to mining and maybe Ethereum in particular because of the Tornado Cash turmoil. I don't see issues with running a Bitcoin or Lightning node on Hetzner, though. To be clear, I'm not affiliated in any way with Hetzner.
A non-trivial number of Lightning nodes seems to be hosted at Hetzner according to figures at e.g.
https://mempool.space/graphs/lightning/nodes-per-isp.
8.3. The transmission of spam mail is prohibited. This includes in particular the sending of unauthorized, unsolicited advertising to third parties. When sending emails, it is also prohibited to provide false sender data or to disguise the identity of the sender in any other way. The operation of applications for mining cryptocurrencies remains prohibited. These include, but are not limited to, mining, farming and plotting of cryptocurrencies. We are entitled to lock the Customer’s access to their Hetzner services or account in the event of non-compliance.
4. Published Content.
We are not obligated to review your content. It is your responsibility to identify the content as your own or as third-party content. You are not allowed to publish content that may violate the rights of third parties or otherwise violate the federal or any state law of the US. You are not allowed to publish content that may violate the rights of individuals or groups of people, or that insults or denigrates these people.
You are not allowed to publish any content that infringes upon the rights of third parties or otherwise violates the law. This includes, in particular, but is not limited to, pornographic or obscene material, extremist content or content that offends common decency, gambling, and material that could seriously endanger the morals of children or young people; this also includes the publication of defamatory content, insults or disparagement of persons or groups of persons. Furthermore, the operation of applications for mining cryptocurrencies is prohibited. This includes, but is not limited to, mining, farming and plotting of cryptocurrencies.
In the case of non-compliance, we are entitled to lock your access to the Service and/or to your account.
Dedicated Server Service Agreement
We strive to keep our networks operating at the highest possible level, so all of our clients benefit from it. Therefore the following actions are prohibited:
- Operating applications that are used to mine crypto currencies
- The scanning of foreign networks or foreign IP addresses
- Manually changing the hardware address (MAC)
- The use of fake source IPs.
The same "operation of applications for mining crypto currencies is prohibited" is repeated at a few more spots at Hetzner's legal statements. I interpret this that you can ignore this in the context of operating a Lighning node in particular.