I just wanted to share my experiences of visiting Taiwan as a dual British/Taiwanese. I was born in Taiwan, left Taiwan when I was 8, gained Infinite Leave to Remain (ILR - I think the equivalent in most other places would be called permanent residence) at 18, and naturalised as a British citizen around 20/21. I have a Taiwanese ID number in my passport, a stamp in the remarks section of my passport that says I require permission to exit Taiwan, and an overseas Taiwanese status stamp.
I went back for a week last week and this is what I did and what people told me:
1. I gave my British passport number to the airline and I left Britain on my UK passport.
2. I gave both passports to the border control on entry and the guy took my Taiwanese passport, and said that I should always enter Taiwan on a Taiwanese passport, I asked about the stamp in the remarks section of my passport about exit permission, and told him that I would want to leave in a week and he gave me a list of like 5 things I could do - I didn't really understand all the options, but took away that one option was just to talk to the immigration office, I think another was to talk to an army office.
3. My Mandarian isn't fantastic, so I called the foreigner helpline (0800-024-111) and asked about this, they pointed me to Taipei City Service Centre line for "Foreign Student、Draftee Consulting Hotline" (02-23883929). I called them and they told me that I could apply for exit permission of any of the service centres and that it would take about 30 minutes. The list is at http://www.immigration.gov.tw/ct.asp?xItem=1092844&ctNode=32289&mp=2
4. I took a taxi to the Taipei Service Centre, it's at "No.15, Guangzhou St., Zhongzheng Dist., Taipei City" (or 臺北市中正區廣州街15號 in Chinese), the page for the centre is at http://iff.immigration.gov.tw/ct.asp?xItem=1085123&CtNode=29682&mp=S002. Once there, I explained to guy there what the border control and the hotline guys told me, and they said it wouldn't take 30 minutes. He asked me to photocopy my passport details page, and the page with my overseas Taiwanese status, filled out some paperwork and asked me to sign it (I can't write Chinese - I suspect anyone who can would have to fill out stuff themselves). He also wanted to know my address in Taiwan, which I gave the hotel address that I was staying in. After that, he stamped the back of my passport in the remarks section, there are block divided pages, and he stamped a date in one of those blocks. He also told me that I would be able to enter and exit up until the date listed on the stamp, without coming back to the office every time I left Taiwan (although the 183 day rule still applies - see other threads about this). He also told me that I should always arrive in Taiwan under my Taiwanese passport - I told him that I had given both to border control and they took the Taiwanese one.
Edit: This whole experience took about 15 minutes or so, excluding the taxi ride - there wasn't even a queue when I arrived at the office.
5. Checked in on the return flight, giving British passport number to the airline.
6. I left Taiwan on my Taiwanese passport, the border control officials did check the remarks section for whether I needed permission to exit, and for the exit permission stamp.
7. Arrived in the UK and entered on my British passport.
I hope other people find this useful.