Samsung Galaxy Note 3 : Tried to install TWRP, can't boot into recovery mode now I followed the instructions for the Odin install method here and and installed the twrp-3.2.3-0-ha3g.img.tar using Odin (Reboot in Odin was unticked).

Everything went fine, but when I had to launch recovery mode by holding down Volume Up + Home, and the Power button (which is the combination to boot to recovery mode) it just booted to Download mode (which is accessed by holding down the Volume Down key instead of the Volume Up)

The TWRP site says:


  Note many devices will replace your custom recovery automatically during first boot. To prevent this, use Google to find the proper key combo to enter recovery.


But I haven't been able to find anything that would indicate the key combination to have changed.

Of note is the fact that the normal Download Mode boot has the Warning screen where you have to press Volume Up to continue and says:

ODIN MODE
PRODUCT NAME: SM-N9005
CURRENT BINARY: Custom
SYSTEM STATUS: Custom
REACTIVATION LOCK: OFF
KNOX WARRANTY VOID: 0x1 (4)
QUALCOMM SECUREBOOT: ENABLE (CSB)
RP SWREV: S2, T2, R2, A3, P3
SECURE DOWNLOAD : ENABLE
UDC START


and then just the "Downloading... Do not turn off target !!"

Though when I use the Recovery Mode Boot it skips the Warning screen and it adds this line on top of all the previous lines:

Could not do normal boot.


It can still boot to the normal Android

I'm using a Samsung Galaxy Note 3, Model Number: SM-N905, Android Version 5.0
I had the same problem like you before. But I figured out the solution. We both downloaded and flashed the wrong TWRP version for Note 3 N9005: ha3g.img.tar. The correct one is hlte.img.tar which you can download on this link: https://dl.twrp.me/hlte/ 

I hope it helps you :)