When do I use 儿 to modify a word? I've noticed that for words like 饭馆 (fànguǎn), 一点 (yīdiǎn)，and 玩 (wán) natives add 儿 (er) to the end while speaking. 

A Wikipedia article about the character says that the phenomenon is called "r-coloring" and that it is dialectical for Beijing and other parts of Mainland China. My Mandarin teacher, who is a Beijing native, insists that we use it while speaking and writing.

Outside of Beijing (and my classroom), how commonly is it used? Should I use it in writing as well?
For the standard mandarin that is regulated by mainland China, the "r-coloring" is part of it. There is an official test for speaking mandarin, "r-coloring" is mentioned in its guideline(《普通话水平测试(PSC)大纲》) and must be tested:


  2·2 读双音节词语50个。
        目的：除考察应试人声母、韵母和声调的发音外，还要考察上声变调、儿化韵和轻声的读音。 
  
  2.2 Read 50 two syllable words. Tone sandhi, "r-colored" final, and neutral tone should be tested, aside from testing how well one pronounces each initial, final, and tone.


About when should one read or write as "r-colored", let me quote something form 《现代汉语词典》, which is explaining the notations within the dictionary:


  6 书面上有时儿化有时不儿化，口语里必须儿化的词，自成条目，如：【今 儿 】、【小孩 儿】。书面上一般不儿化，但口语里一般儿化的，在释义前加‘（~ 儿）’，如【米粒】条。 ....
  
  When a word is sometimes written with "儿" and sometimes not, but must be spoken as "r-colored", it would have its one entry, e.g. 【今 儿 】、【小孩 儿】。When a word is usually not written with "儿", but usually spoken as "r-colored" in speech, ‘（~ 儿）’ will be appended to explanation, e.g. 【米粒】


(Note, the dictionary used a small font for the above italic 儿 in the quote. I don't know how to type it here)

The dictionary's notation may be irrelevant here, but we can see there are words in standard mandarin that written “儿” is optional, or usually not written with "儿", but should be spoken as "r-colored". Examples are given in the quote. The written form and the spoken form sometimes do not match.

About how it is applied in real life, you should remember that most people in China do not speak standard mandarin (very well). The application of "r-colored" tones is quite regional and may differ a lot from place to place. Some people do not use them at all, and some use a lot. I think you should follow your teacher as when to use "r-colored" tones in speech.

As when to use it in writing, though the dictionary listed a lot of words as "sometimes written with 儿 and sometimes not", there is, in fact, a small set of words that are usually written with "儿". You can drop all "儿" in writing with words outside the set. I cannot list all words in the set. The most common one that usually written with "儿" might be "一会儿". You can follow your reading materials as when to "儿" in written form.