If I have a LN node without open channels, does that improve the robustness of the network? I have a BTC and LN node running 24/7 on a Raspberry Pi 3, and no BTC at the moment to open a channel.
Running a Lightning Network node without any open channels doesn't really make the Lightning Network more robust. Unlike the main Bitcoin layer, the Lightning Network does not rely on or require other nodes to verify transactions. Thus a Lightning Network node without any open channels is simply a data relayer which there are many (all of the other LN nodes). There really isn't much data to relay anyways, and you relayhing it does not ultimately help LN that much.
The usefulness of a LN node is to provide capacity for routing payments over additional payment channels. Since you do not have any payment channels, your node isn't really helping with that.