how to put a list into viewstate

Refer - Pushing a Geneic List of <t> to View state[^].

Quote:

Here's another quick way if you just wanted to Save things to the view state:

1. Mark the Class (and all parent classes if any) whose List you want to store in the ViewState as [Serializable].
2. Convert the List to an Array.
3. Store in ViewState.

But the ListItem you have used is not Serializable.
So, create your own class with public key/value properties, mark it Serializable and convert to Array and store in ViewState (as described in that link for Customer Class).

Note
Store List in ViewState if you have less number of Data in List.