Try using the event API. See Event API (HSSF only) and XSSF and SAX (Event API) in the POI documentation for details. A couple of quotes from that page:
HSSF:
The event API is newer than the User API. It is intended for intermediate developers who are willing to learn a little bit of the low level API structures. Its relatively simple to use, but requires a basic understanding of the parts of an Excel file (or willingness to learn). The advantage provided is that you can read an XLS with a relatively small memory footprint.
XSSF:
If memory footprint is an issue, then for XSSF, you can get at the underlying XML data, and process it yourself. This is intended for intermediate developers who are willing to learn a little bit of low level structure of .xlsx files, and who are happy processing XML in java. Its relatively simple to use, but requires a basic understanding of the file structure. The advantage provided is that you can read a XLSX file with a relatively small memory footprint.
For output, one possible approach is described in the blog post Streaming xlsx files. (Basically, use XSSF to generate a container XML file, then stream the actual content as plain text into the appropriate xml part of the xlsx zip archive.)