Would Tally.ERP 9 64-bit application resolve Out of Memory issues?

Yes, Tally.ERP 9 would resolve the out of memory issues.

Today, out of memory issues in Tally.ERP 9 arises of multiple reasons:

Due to volume of data (masters etc.)

Volume of entries (large number of Ledger entries/bills etc. in a single Voucher being edited etc.)

Non-Optimized code (Default / STAT or customized TDLs)