I am going to point you to two similar questions I answered last week:
Are Oracle 10g and VB 6.0 compatible?
Impact of Oracle 9i upgrade on Java front end
The only difference between your question and the previous one is that they are using VB 6, whereas your applications were created in Delphi.
It would be nearly impossible for me to predict what types of application issues you will encounter when you upgrade to Oracle 10g. I have no information about your application, the data or how it's used. The common theme (and the message I keep repeating to others with this type of question) across these questions is that you must test your upgrade in a non-production environment. The testing must be documented, thorough, and involve your application support folks as well as business users. As well, the upgrade process (especially one like this that is taking a database from Oracle8i to 10g) must be repeated several times, each time with extensive testing. Time spent testing is never wasted!
Dig Deeper on Oracle database export, import and migration
Related Q&A from Maria Anderson
I have been trying to install Oracle 8.1.7 on SUSE Linux 9.0 and got the error: "Error in invoking target install of makefile /opt/oracle/...../*.mk." Continue Reading
I'm getting errors while installing the Database Configuration Assistant. Continue Reading
Can I use /var/opt/oracle/oratab to specify listener information? Continue Reading