In transcribing the word as [fij], you are committed to saying that this is a closed syllable (unless you invoke dubious abstract structures where j is not syllabified, or is the onset of a syllable containing no vowel). The letter [j] is not a vowel, and under normal assumptions about syllabification, it would be part of the syllable defined by the preceding vowel i. That, by definition, makes the syllable closed. You have choices in transcriptions, which do not reflect some immutable phonetic absolute, they reflect phonological analysis. If you want to say that the syllable contains a diphthong, then you should transcribe the word as [fii] (which is different from disyllabic [fi.i]), and if you want to say that the syllable contains a monophthong and is open, then you should transcribe the word as [fi].
You should also look at actual practices for transcribing vowels of English, because the diversity of practices very clearly show how transcriptions are simply conventionalized and cannot be fully rationalized. The word "fee" can be transcribed many ways, for example [fi, fi:, fij, fɪj...] In general, i-ending diphthongs are written with final ɪ,i,j as well as raised versions of any of these letters. One is always free to write "car" as [kaɹ] and call it an open syllable by labeling the sequence a rhotic diphthong.
In this case your teacher presumably gave you a transcription, and wants you to come up with an analysis. The question you might later pose to the teacher is, why that particular transcription? If the answer is "because this is an closed syllable", then the follow-up question would be "How do you know that it's an open syllable?". If s/he thinks it is an open syllable, the the follow-up questio would be "why [fij] and not [fii] or [fi]?"