There are a lot of tips, hints and links for Southeast Asian food on offer in Taipei, but I thought I'd put forth a few suggestions I haven't yet seen knocked about online as my favorite destinations for food from the peninsula and islands.
Gongguan Night Market, near the Molly's Used Books behind Taipower Building. From Gongguan Night Market, turn in the alley near the Vietnamese Restaurant and head to the end, where you come across a small park and the end of the night market. It's on the road on the far side of the park, between a Korean BBQ and a coffeeshop.
Yangon looks like a Thai restaurant that happens to have a Burmese name - no, I can't figure out if I should call it Myanmar or Burma - and if you order incorrectly from the menu, you'll get just that, Thai food. But order correctly, or better yet, compliment the owners by specifically requesting Burmese food recommendations, and you are in for a real treat. We ordered three dishes and a green papaya salad. The eggplant dish felt very Chinese, with sweet soy sauce and a flavor reminiscent of Yunnan cuisine. The meat-in-a-stone-bowl was curried, with flavors from northeast India. The papaya salad and shrimp fritters (I know, I know, but I LOVE shrimp fritters) were very Thai. Put them all together and you have Rangoonian bliss. Also, very affordable.
South East Asia Food Center Xinyi (all kinds)
(at least that's what I think it's called - I've lost the card)
Near the International Trade Building with all the consulates in it (that tall square building between the Grand Hyatt and the World Trade Center) - cross Keelung Road and head slightly to the right. It's the first lane to the left of a place offering Singaporean food, which we haven't had the pleasure of trying yet. Walk down the lane a bit and it's on the right.
The owner, whom I believe is named Winston, is Vietnamese but the place offers food from all over the peninsula. He speaks great English, and the place is packed with Taiwanese office workers coming for a good-value lunch in Xinyi, who want Southeast Asian food but don't really want to pay Shinkong Mitsukoshi prices for it. They have Singapore noodles, Vietnamese pho and spring rolls, green papaya salad, curry fried rice, laksa, Thai curries and more...all for excellent prices.
The green papaya salad is more Lao in flavor than Thai, so those used to the hot, sour Thai style and unfamiliar with the more coriander-and-oniony, crunchy, lemony Lao style might be surprised.
Shida Night Market, at the very end - turn in the road that begins at the Fubon Bank (Shida branch) on Heping E. Road and it's visible on the left
Not exactly Indonesian food, but good. They do not do Padang-style 'small dishes', something I remember quite fondly in Sumatra when we gorged ourselves silly on Padang food in Padang itself...but what they do offer is quite nice. Be sure to request 'extra spicy' or 'local style' - the chef is Indonesian and can cook it up for you the way the staff would eat it, but if you don't say anything you get something a bit milder. They do standard Indonesian fare - nasi goreng, mie goreng, rendang, satay - at bargain-basement prices. I don't think I've ever paid more than NT100 for a meal there. Plus they have a cute white dog named Oliver.
The vendor guy next to him who sells crispy Thai spring rolls on a stick also cooks up a tasty treat.
Pinoy (Filipino) food of all kinds
Sundays on Zhongshan N. Road between Minquan W. Road and Yuanshan MRT stations. Walk up between the two and turn right into the lanes at just about the halfway point. Options abound.
I would make specific recommendations but frankly, pretty much everything is good. Try one of the places that looks like a Taiwanese buffet, but you pay by the small dish of food (1 or 2 per person, go with a group and share or go alone) so just order whatever looks good and, frankly, it probably is. If you have no stomach for innards, stay away from the sisig. I could handle sisig in the Philippines because it's soft and tender and bears no trace of its, um, gutsy origins, but the sisig in Taipei is a little more blatant in advertising exactly what it is and where it comes from. This is food for Filipinos in Taiwan on their way home from church on Sunday, so you know you're getting the real thing.
New Bangkok Restaurant (Thai)
Easily found in a lane on the eastern side of Fuxing N. Road between Zhongxiao Fuxing MRT and Breeze Center.
Their fried eggplant and shrimp fritters left something to be desired, but it's worth it to go for the amazing minced basil chicken and green papaya salad, which is among the best I've had in Taiwan. Its hot, sour, sweet and savory flavors are perfectly balanced to create a heavenly chord, like the end of a good Bach fugue, on your tongue.
Thai, Yunnan and Myanmar Food (Neihu branch ONLY) - Thai
Ruiguang Road, Neihu, across the street from the large bus stop of the same name, near E-Ten's office and the Barista Coffee - incidentally the onion pancake guy next to that Barista does an awesome pancake.
Other branches of this restaurant have disappointed me with lackluster tea and mediocre food, but this branch does something very right. I've always been happy with everything I've eaten here, including the soft tofu in coconut sauce, the red curry chicken, the green papaya salad, the greens with sliced pork, and, well, everything.
Tiny Vietnamese pho stall on Heping W. Road (Vietnamese)
Basically if you head west on Heping W. Road from Roosevelt Road and just continue on for about ten minutes not too far before the pedestrian overpass directly before the botanical gardens and old Academy of Science Building (as well as the other historic buildings surrounding it), and it's on your right in a barely noticeable little card-table-and-white-wall storefront.
I can't remember the name of this place, but the pho is so good and so authentic that it's worth a mention. Really. If you are in that area, maybe heading to the botanical gardens or bird market, it's worth planning a lunch or dinner here if you are a pho fan. The owners are a very friendly Vietnamese couple who are delighted to hear that their food is excellent, and a steady procession of overweight dogs from the next store over comes in as you eat (this is more adorable than it sounds). Really, it's good. Forget Madame Jill's or Yongkang Street and head straight here.
Pho stalls in Xindian and Tonghua Night Markets (Vietnamese)
There's one on the righthand side of the road in Xindian, not far from the pedestrian bridge and partially hidden by some metal fencing. The other good one is in Tonghua Night market about halfway in, down one of the small lanes lined with food stalls (righthand side lane if you enter from Keelung Road, righthand side stall). The one in Xindian makes excellent pho with loads of basil and the other has delicious fresh spring rolls with large whole shrimp for a steal.
If in Xindian, start with a bowl of pho here and head to Athula's on the other side of the pedestrian bridge entrance for a curried meat roll.
Fried Banana and Thai Iced Tea stall (Thai)
Raohe Night Market, near the far end if coming from Houshanpi MRT...opposite end from the temple, but near the bus stop with buses down Nanjing E. Road
This is really just a tiny stall that sells fried banana crepes with a choice of topping - honey, condensed milk or chocolate - and Thai iced tea. It's on the righthand side if you begin at the temple/Wufenpu Fashion Street end of the market and almost at the opposite end. A perfect ending to a meal at Ala-din (delicious, spicy Pakistani food with unlimited vegetables and naan - the veggies are fried in real ghee, not the crappy vegetable oil substitute one so often sees), also in Raohe Night Market.