When you create an invoice, it reveals your node's public key, which stays the same for the lifetime of your node. This has privacy implications:
if you post invoices in public places, the pubkey may link together your identities across platforms
the recipient of an invoice can link multiple invoices from the same node to eachother
anyone can view the value of bitcoins that a node pubkey has in non-private channels
anyone can view the channel opening and closing history of a node pubkey
anyone can match public channel opening and closing events to the on-chain inputs used to find them, and can trace those inputs using blockchain analysis
if your node is configured to broadcast a public IP address,
your invoice reveals the node's IP address and approximate geolocation
network participants can link together multiple nodes hosted at a single IP address
if you use multiple nodes with a single identity, invoice recipients can link together those two nodes
When you pay invoices, much less information is revealed:
A payment does not indicate to the recipient what node pubkey it originated from
A node that routes a payment does not know conclusively which node the payment came from or which node the payment is going to
the routing node only knows which node forwarded it the payment and which node it was instructed to forward the payment to
If the recipient service uses accounts of embeds user-provided information in the invoice, then obviously the service can link together invoices.
What things stay private?
The existence and balance of private channels remains secret to every node except for the node you have opened the private channel to, unless that node chooses to disclose that information
Your on-chain wallet balance that is not contained in a channel is not linked to your node pubkey, unless though it may be possible to link on-chain wallet balance to previously closed public channels through blockchain analysis
If you use separate nodes with separate identities, there is not a straightforward way to link them
How does using Tor with Lightning improve your privacy?
By connecting your node to Tor, other Lightning nodes connected to Tor can open channels to your node without requiring you to broadcast a public IP address
The current Lightning Network specification includes a solution to mask routing data from all intermediaries, based on Sphinx.
On Lightning, the payer determines a path over the peer-to-peer network and wraps a payment package in layers of encryption. Apart from just relay information, each intermediary also unpacks some additional data. This includes amounts, fees and more, along with allowing all intermediaries to set up a step in the payment chain.
Importantly, all intermediaries only learn from which channel they receive tokens, and to which channel they must forward the payment. The intermediaries have no idea whether they are the first step in the chain, the last step, a step somewhere in the middle, or perhaps even the only step. Whoever originally sent the transaction, and the one who ultimately receives it, remain known to only the sender and the receiver.