I can't say that I will ever have the impulse to burn any of my journal works. I write for various reasons, but one thing that always is a part of it, it to learn. That is really the only reason I consistently journal. Now, I understand Browning's concern of privacy, REALLY I DO, but in my opinion, just be careful.
In terms of burning away the past, I understand that too. There are times I look back at things I've written and cringe at (Depending on the day it may be the content of my terrible use of grammar), but reading those things remind me of life lessons and events whether good or bad. It really helps me to understand everything I've been through to make me, me.
I think the prospect of burning journals is an interesting one. Like I said already, I have no interest. However, I think the benefits of it could vary depending on why one journals. For me, I focus one events (mainly good ones) and I focus on thoughts I have, although usually random and pointless, that are really intriguing to go back to years later. For someone who uses journaling in a therapeutic way, the burning seems liberating. It would be nice to watch your bad memories be put behind you.
Journaling for me is fun. Its something I do for pure enjoyment so I have no interest in destroying those works.