Teaching English as a foreign language (TEFL) refers to teaching English to students whose first language is not English. So usually EFL refers to teaching English is a second language. In English speaking countries, the preferred term is ESL.
Teaching English (or any other language) as a third language usually refers to teaching the respective language to either bilingual students or just students who already have a certain degree of proficiency in a second language.
Check out some interesting articles of L3 teaching:
English as a second language (ESL) is often used for non-native English speakers learning English in a country where English is commonly spoken. English as a foreign language (EFL) is used for non-native English speakers learning English in a country where English is not commonly spoken. Teaching English as a third language usually refers to teaching English to bilingual students. For example, in the Basque Country, there are two official languages, Basque and Spanish, and English is taught as a third language. The acquisition of an L3 shares many characteristics with the acquisition of an L2 or EFL but it also presents differences, such as learners' context, learners' needs, teaching method, etc.
I suppose that learning English as L3 shares many characteristics with learning English as a foreign language. However, there can be certain differences stipulated by the background of learners.
My mother tongue is Russian, the second language that I learned is Hebrew, and English is my third Language. In my opinion, learning the English as third language is more simple, since the skill of improving the vocabulary is already very developed and fear to talk on the foreign language is already minimal.
I am a researcher in second language education, and l have learned English as a second as well as a foreign language. I have also taught English as a second and foreign language (ESL and EFL). Now, I am making effort to learn Spanish as a third language.
From my personal learning, research, and teaching experience. I believe that language contexts matter tremendously as to the teaching and learning strategies, particularly whether or not the target language is a medium for instruction, for instance EFL and ESL. Thus I tend to believe that learning English as a third language (L3) would be similar to English as foreign language (EFL) if the instructional language is other language rather than English and if the students have similar exposure to English in both environments. Of course the goal of the curriculum would play a significant role as well.
You may find very useful a discussion on L1, L2, etc. offered in the first chapter of the book "Fundamental concepts of language teaching" written by Stern, H. H.
This is an excellent book and the best thing is that it is available at the University of Malaya library!