I think you messed up some things in your question. Lucene (I know nothing about Lucene,NET, but I suppose is the same) is a library used to analyze, split in tokens, and store documents in order to be able to query and retrieve them later. Lucene has a pretty old but effective model, it uses inverted trees to find and retrieve documents. Without further details, all documents are split in tokens (terms), and for each term is maintained a data structure, which stores all the documents which contains the given term. As a data structure could be used a BTree, a hash table and in the latest major revisions you can even plug in your own data structures.
A BTree (see Wikipedia page for further details), is a kind of a tree data structure, which is appropriate for working with big chunks of data and is often used for storing tree-like ordered structures on disk. For in-memory other trees performs better.
Murmur hash (see Wikipedia page for further details), is a family of hash functions used in hash table. The implementation of the hash table is not important, it could be a standard chained implementation or more advanced open hash addressing scheme. The idea is that the hash tables allows one to get fast a key, from an unordered set of keys, and can answer to tasks like: is this key part of this set of keys? which is the value associated with this key?
Now back to your main problem. You have one library (Lucene) and to data structures, both data structures are used in Lucene. Now you see that it is not possible to answer your question in these terms since they are not comparable.
However, regarding you footprint and performance part of the question. First of all you have to know which kind of operations you need to implement.
Do you need only get value for key, or do you need to find all elements in a range? In other words do you need order or not? If you do, than a tree can help. If you do not, than a hash table, which is faster could be used instead.
Do you have a lot of data which does not fit the memory? If yes than a disk-based solution would help (like BTree). If your data fit the memory, than use the fastest in-memory solution and use disk only as a storage (with a different structure, much simpler).