Why aren't soldering tips coated with platinum?
Sigh - as pointed out by @user83678 some 3+ years after first writing this, I answered for Pd, not Pt. That will teach me to pay attention. But, the general answer remains the same. There are enough interactions of platinum with either lead, tin, or silver (a common component in lead-free solders) such that a platinum tip does not make metallurgical sense. 

Lets look first at Pt-Pb, with the CALPHAD parameters from Z.H. Long et al., J. Phase Equil. Diff 30 318-322 (2009):

 

While Pt does not have a high solubility in Pt, the presence of multiple intermetallics indicates problems: either Pb reacts with the Pt time, or Pt dissolves into the solder melt and forms an intermetallic there. Neither is desired. 

For Pt-Sn, one gets (with CALPHAD parameters from V. Grolier and R. Schmid-Fetzer, J. Alloys and Compounds 450 264-271 (2008)):


Here the Pt happily sucks up lots of Sn, and then there are all those intermetallics. 

For Pt-Ag (from I. Karakaya and W.T. Thompson, Bull. Alloys Phase Diagrams 8(4) 334-340 (1987)) one gets:



The fcc Ag and Pt have large solubilities for each other (but not complete solid solubility, somewhat surprisingly). Again, you get lots of interactions between a Pt tip and a lead-free solder containing Ag. 



Aside from my comment that it is not hard to keep a soldering tip in tip-top shape, I will point out several materials issues. 

For this purpose, I will stick to the interactions of Pd with classic Pb-Sn solder. A peek at the Pb-Pd phase diagram (ASM International, or G. Ghosh, Met. Trans. 30A, 5-18 (1999)) one readily sees that palladium has a large solubility of lead, more than 10% at room temperature. Since the solubility rises with temperature, cooling a Pb-saturated Pd tip from soldering temperature to room temperature would result in Pb precipitates forming, likely disrupting the mechanical stability of the Pd tip.

A glance at the Pd-Sn phase diagram (same paper referenced above) shows a fairly similar situation (not so surprising since Sn and Pb are in the same column, but that isn't a given in phase diagrams). Again, even at room temperature the solubility of Sn in Pd is in excess of 10%. 

For both systems there is a binary phase, Pd$_{3}$Sn or Pd$_{3}$Pb, very close to the fcc palladium phase field, as well as a range of other phases stretching to higher Pb or Sn contents. The ternary diagrams included in the Ghosh paper show many phases present. This means that reactions between molten Pb-Sn solder and a platinum tip won't stop at just saturating the fcc Pd with Pb or Sn - the reaction will continue, making compounds across the phase diagram and rapidly ruining your fancy Pd tip.

For lead-free solders, note that Sn remains a primary component in many of them, so the same things apply.  

I have included a (calculated from the CALPHAD parameters in Ghosh's paper) the Pb-Pd phase diagram.