This is a pretty broad-based question as it has less to do with what is being printed versus what type of printing is being used to make the posters and banners, although material will also have somewhat to do with the type of imprinting being done.
With poster print processing, the type of inks being used are potentially several different types, again, depending on the printer itself. If you need a short run of posters, the most cost effective way will be is to have a digital company print you posters on a roll-to-roll wide format digital print machine. The inks today, in the US, are soy-based eco-friendly inks, although you'd be hard-pressed to tell the difference between solvent and soy base inks, both by the smell and by the quality of the end product.
Inks in Screen Printing
Posters could also be screen-printed, although this method is not as common as it once was. Screen imprinting inks are typically soy-based also in the US (it's the law), although there are a few types of ink within this kind of genre. Most commonly used would be UV, or ultraviolet light cured inks. When doing the four color process (4 CP) posters, the inks are laid down successively, first yellow, then magenta, then cyan, and finally, black. You'll often see these ink colors annotated as C-M-Y-K, which are the four colors of the 4 CP process.
With screen printing, the inks have to dry between colors printed, so the UV process is really the only way to go in the modern era, though we're seeing the digital process supplanting this process except on lower color quantity, higher quantity prints and believe that this will be obsoleted within a few years, although we could be wrong in this assessment. Inline screen presses with various color screens can still turn out very high quantities of posters or banners even printing 4 or 6 colors, but the fastest one is the offset lithographic, more commonly called just offset printing.
Inks in Offset Printing
Again, the inks used with offset printing, by law in the US, are soy-based, but have small differences in the formulation, but the outcome is essentially the same. There are some offset printers that print very small items such business cards (although they are "gang-printed" on sheets and then sheered, typically. There are larger format offset printers that can print large posters as well. This print process, once set up, is used for very large runs of posters. While banners can be offset print, there are size limitations that will make it so large banners are not likely to be run on an offset press. That domain is generally reserved for digital printing now, where printers can print over 16 feet in width and as long as the roll is (typically 50 yards).
So while ink will vary as far as formulations for the type of printing done, there may be only an added solvent that will make a vinyl or polypropylene banner printable that isn't in the paper poster ink, but the differences are minimal, and most inks now are UV curable or quick drying as they come off a heater printer platen, not the slower drying inks that were around when I began in the printing business.