Does a gene that regulates itself necessarily be a transcription factor?
First of all you need to clarify what exactly is the output that needs to be regulated. We often say regulation of expression of a gene but what expression ultimately leads to is the activity of the gene (whatever it might be: enzyme based catalysis, cytoskeletal rearrangement, response to an extracellular signal). Activity of course depends on the expression level of the gene but it also depends on other factors such as covalent modifications, interactions with other genes (like dimerization) etc. Moreover, for genes which code for proteins, expression also has two stages. 

Coming back to your question:


  If a gene is NOT a TF gene, can it regulate itself?


Yes it can. There are many modes of gene activity regulation as I already mentioned. Examples where a non-TF gene can regulate itself:


Feedbacks in cell signalling which happen via activity modulation by protein phosphorylation [1].
Regulation via post-transcriptional regulation via RNA binding proteins (RBP) or regulatory RNAs such as miRNA. MAPK signalling pathway has a feedback via the RBP, Rnc1, which upon phosphorylation binds to and stabilizes the Pmp1 mRNA. Pmp1 is a phosphatase which dephosphorylates Rnc1[2]. This loop contains regulation by both post-translational modification and post-transcriptional mRNA stabilization. There are many other examples involving RBPs. An miRNA, let-7 is post-transcriptionally regulated by Lin28 (promotes degradation) which in-turn is post-transcriptionally repressed by let-7 [3].


There are many examples of metabolic feedbacks too which operate via allosteric regulation of the enzyme activity. 

If you are only interested in regulation of gene expression then you can ignore phosphorylations and allosteric regulation but there are several examples of post-transcriptional regulatory interactions which also contain feedbacks.



References


Dougherty, Michele K., et al. "Regulation of Raf-1 by direct feedback phosphorylation." Molecular Cell 17.2 (2005): 215-224.
Sugiura, Reiko, et al. "Feedback regulation of MAPK signalling by an RNA-binding protein." Nature 424.6951 (2003): 961-965.
Rybak, Agnieszka, et al. "A feedback loop comprising lin-28 and let-7 controls pre-let-7 maturation during neural stem-cell commitment." Nature Cell Biology 10.8 (2008): 987-993.