Here’s a masterpost of resources to aid in learning South Asian languages! This is by NO means comprehensive. A very limited number of languages are represented here. This search was really frustrating because the more I looked, the more I realized how many dozens of South Asian languages are simply not represented or their existence acknowledged. If you have any resources/languages you’d like to me to add, please let me know! 

Assamese
iLanguage

Balochi
Balochi Basics
UChicago Resource List

Bengali
Everyday Language Learner
King’s College Resources
Bengali Masterlist

Brahui
Overview
Common Phrases (1)
Common Phrases (2)

Burmese
Cornell Burmese Resources
University of London Resources

Bumthang
Grammar (1)
Grammar (2) 

Chakma
Mastersite

Dari
King’s College Resources

Dhivehi
General Overview
Primary Grade Resources

Dzongkha
Vocabulary
Grammar

Dimasa
Grammar

Dogri
Overview

Halbi
Script/Language

Hindi/Urdu
WUSTL Hindi-Urdu Resources 
HindiUrdu.net
Hindi-Urdu Flagship Resources
Columbia Hindi-Urdu Resources

Hindko
Overview

Ho
Swarthmore Resources

Farsi
Virtual Persian
Resource Masterlist (1)
Resource Masterlist (2)

Garo
Overview

Gondi
Mastersite

Gujarati
King’s College Resources
Language Reef
Gujarati Learner
Gujarati Masterlist

Gurung
Video Series

Jingpho
Grammar

Kannada
UofIowa Resources
Language Reef

Kashmiri
Mastersite

Khams Tibetan
Language Materials

Kharia
Kharia-English Dictionary 

Khasi
Learning Basic Khasi

Kurukh
Kurukh Grammar

Kokborok
Learn Kokborok

Konkani 
Language Reef
Literature Resources
Common Phrases 

Lepcha
Grammar (1)
Grammar (2)
Basic Phrases

Maithili
Maithili Sentence Structure
Background and Linguistics

Malayalam
UofIowa Resources
Common School Phrases

Manipuri 
Common Phrases 
Language Reef

Marathi
iLanguage

Mishing
Grammar

Nepali
Nepali: Beginner’s Primer
NepalGo
NepaliLanguage.org
UofIowa Resources
Nepali Masterlist (1)
Nepali Masterlist + MP3 Lessons (2)

Nyishi
Grammar

Odia
Resource Masterlist 

Pashto
King’s College Resources

Punjabi
LearnPunjabi.org
King’s College Resources

Shina
Grammar Overview
Grammar and Vocab

Sindhi
Sindhi Masterlist

Sinhala 
Sinhala Basic Course 
AISLS Sinhala Studies
Sinhala/Tamil Dictionary

Sora
Swarthmore Talking Dictionary

Tamang
Writing Tamang

Tamil 
Tamil Reader/Grammar
AISLS Tamil Resources

Telugu
Learning Telugu
UofIowa Resources
Columbia Resources

Tibetan
Emory Resources
Tibetan Masterpost (1)
Tibetan Masterpost (2)
English-Tibetan Dictionary

Tshangla
Overview

Tulu
Common Phrases (1)
Common Phrases (2)
Common Phrases (3)
Common Phrases (4)

Apps
Mango Languages (Bengali, Dari, Dzongkha, Farsi, Hindi, Malayalam, Pashto, Punjabi, Tamil, Telugu, Urdu)
Livemocha (Farsi, Hindi, Marathi, Urdu) 
Duolingo (Hindi) 

Other Resources
So You Want to Learn a Language? (has resources on a ton of languages!)
Digital South Asia Library
Digital Dictionaries of South Asia (this resource is awesome!)
UChicago Recommended Resources
Omniglot (has some basic info for a lot of languages)
Languages of India Mastersite

Playing in the dust - Shan State - Myanmar by Pascal Boegli
Via Flickr:
Children in Wun Nyat, a Loi village close to China border, Shan State, Myanmar. Loi people live in longhouses where multiple families share the same building.