A glass syringe should not be corroded by the TFA. For the tubing, personally I used PTFE without problems, but another option could be polyethylene, polypropylene or PCTFE. In general oxygen-containing plastics tend to react with TFA .
Is your tubing required to be flexible? I would suggest PTFE or PEEK (polyether ether keytone). PEEK is not very flexible but it is very strong and will alomost never break. PTFE is flexible but not very strong. In your case strength may not matter.
Cold TFA is a good solvent of terephthalic esters, like Polybutylene terephthalate, polyethylene terephthalate. Maybe your tubing is made of one of those polymers. PTFE is certainly a good choice.
TFA react with the piston bottom portion (black portion) only. So simply fill the solution in the syringe make an air gap between the solution and the piston. It will work, if you use the syringe pump vertically (syringe in vertical positioned)..... It works
TFA is one of the few solvents for polyesters, it also dissolves many other commercial polymers. You might use both glass and teflon (PTFE). Depending on the polymer you are using (for example PET), to improve solubility in TFA you may try to cool the solution. At room temperature TFA is mostly present as a dimer, and at lower T (around 0 °C) the equilibrium goes back to the monomer which is more efficient as a solvent. The same procedure is not valid for PEN, that forms a gel in those conditions