What are alternatives to Proof of Work for consensus of a blockchain that doesn't contain anything of value? For example, a distributed blockchain that would keep track of the weather. Anyone could add a weather reading, and several of these would be stored in each block, just like transactions in a cryptocurrency. There is no reward for mining these blocks, so speed of mining wouldn't be a huge concern for inflation, all that would matter is that all the nodes agree.
It seems to me that Proof of Work would work for this, but it would be lots of effort for nodes to get nothing in return, also the benefit of slowing mining isn't really important. Proof of Stake seems to be the next most popular consensus mechanism, but I can't see how that could possible work in this situation, where you have nothing of value to risk.
What other methods of consensus could work for this?
a blockchain that doesn't contain anything of value

Do you mean a blockchain with no associated token? If the data really has no value, no one will be interested in storing it on a blockchain.
If there is a limited amount of writers (i.e. up to ~100 nodes), it is possible to create a consortium network and deploy a blockchain using a Byzantine Fault-Tolerant (BFT) consensus (e.g. Proof of Authority). These blockchains do not require a token/cryptocurrency for the consensus. This is typically referenced to as a permissioned blockchain (as opposed to a permissionless/public blockchain such as Bitcoin or Ethereum).
Permissioned blockchains have different very different tradeoffs compared to permissionless blockchains. The blockchain is controlled entirely by permissioned nodes which are the only ones allowed to perform writes (in a public permissioned blockchain anyone is able to read). This model generally support a much higher throughput (more transactions per second) compared to permissionless consensus. Since nodes are authenticated in BFT protocols, the consensus model is designed to resists against malicious nodes up to a certain threshold (e.g. 1/3 of malicious nodes).
As a side-node, the flowchart in Fig. 1 from Do you need a Blockchain? can help to determine if a Blockchain is the appropriate solution for a problem.