I've recently bumped into this issue and it was very tricky to resolve. It looks like there was an undocumented change to the drivers used in the SQL 2016 RTM release, as I didn't strike these issues on the same machine when it had the RC3 build.
Anyway in case it helps, here are the setup steps I'm using (with thanks to Dan English for some tips):
Install Oracle drivers: Oracle Client 12c (32-bit) plus ODAC.
a. Download and unzip the following files from http://www.oracle.com/technetwork/database/enterprise-edition/downloads/database12c-win64-download-2297732.html and http://www.oracle.com/technetwork/database/windows/downloads/utilsoft-087491.html ):
i. winnt_12102_client32.zip
ii. ODAC112040Xcopy_32bit.zip
b. Run winnt_12102_client32\client32\setup.exe. For the Installation Type, choose Admin. For the installation location enter C:\Oracle\Oracle12. Accept other defaults.
c. Start a Command Prompt “As Administrator” and change directory (cd) to your ODAC112040Xcopy_32bit folder.
d. Enter the command: install.bat all C:\Oracle\Oracle12 odac
e. Copy the tnsnames.ora file from another machine to these folders:
i. C:\Oracle\Oracle12\network\admin
ii. C:\Oracle\Oracle12\product\12.1.0\client_1\network\admin
Install Oracle drivers: Oracle Client 12c (64-bit) plus ODAC.
a. Download and unzip the following files from http://www.oracle.com/technetwork/database/enterprise-edition/downloads/database12c-win64-download-2297732.html and http://www.oracle.com/technetwork/database/windows/downloads/index-090165.html ):
i. winx64_12102_client.zip
ii. ODAC121024Xcopy_x64.zip
b. Run winx64_12102_client\client\setup.exe. For the Installation Type, choose Admin. For the installation location enter C:\Oracle\Oracle12_x64. Accept other defaults.
c. Start a Command Prompt “As Administrator” and change directory (cd) to the C:\Software\Oracle Client\ODAC121024Xcopy_x64 folder.
d. Enter the command: install.bat all C:\Oracle\Oracle12_x64 odac
e. Copy the tnsnames.ora file from the K drive folder (see 4.a. above) to these folders:
i. C:\Oracle\Oracle12_x64\network\admin
ii. C:\Oracle\Oracle12_x64\product\12.1.0\client_1\network\admin
After all that, most SSRS components worked OK. The exception was the SSRS Portal where Data Source setup and Test worked OK, but (bizarely) running reports gives an error:
ORA-12154: TNS:could not resolve the connect identifier specified
Eventually I gave up on this and went with an EZCONNECT connection string: host:port/sid. I never liked TNSNAMES files anyway, so this was my excuse to kill them for good.
Best Answer
You need to use Oracle's 32-bit Data Access Components for SSDT / Visual Studio development with Oracle data sources:
http://www.oracle.com/technetwork/topics/dotnet/utilsoft-086879.html
Please note if you are deploying to a 64 bit SSRS Server, you need to install the 64 bit version of the components on your SSRS Server.