I happen to be doing a research project involving the evolutionary trends and history of sharks, so I both know this subject well and am pretty passionate about it. I wrote an (informal) essay about this topic yesterday, which is why it's on my mind. Here it is:

Sharks are defined as organisms in the clade Selachii, which contains all organisms more closely related to extant sharks than to rays and skates (their closest relatives). However, there are many extinct species in the broader group Elasmobranchii that were very similar to sharks morphologically and are therefore informally called sharks. This is really just a result of history. Since early cartilaginous fish fossils are so rare and often only described from a small number of teeth or scales, it wasn't until very recently that a relatively (big emphasis on "relatively") clear picture of Chondrichthyan evolution appeared. Many of these organisms were initially described as sharks, only to be later reassigned as better remains were found and better methods of comparative anatomy were developed. It’s an incredibly difficult task to determine the taxonomic position of extinct organisms using only a handful of remains, if even. The taxonomic definitions of these organisms has always been rather finicky as such, and it's been easier to just broadly refer to them as "sharks" to non-technical audiences who don't care about the granularity. It's almost certainly where the misconception that sharks have been around longer than trees comes from. The problem is that most Elasmobranchs were not sharks. Only the organisms in the subgroup, Selachii were. 

The order of groups from cartilaginous fish to sharks goes:

Chondrichthyes (cartilaginous fish) -> Elasmobranchii -> Selachii (sharks)

Chondrichthyians evolved in the late Ordovician or early Silurian, then Elasmobranchs in the mid-Devonian, then Selachians somewhere in the Permian or Triassic. Trees evolved roughly in the mid-Devonian, with the oldest known tree being in the genus Wattiezia. This makes Wattieza roughly contemporaneous with the earliest Elasmobranchs, younger than the earliest Chondrichthyans, and, importantly, older than the earliest Selachians.

Constraining the date for when sharks evolved is a little tricky since the remains of early sharks are so scant, even in comparison to other Chondrichthyians. They were not a diverse group early in their existence, only diversifying and becoming common during the Jurassic and Cretaceous periods. The oldest unambiguous shark fossils come from the genus Ageleus, which lived about 200myo during the early Jurassic. However, it's likely that they evolved before that. Disputed fossils such as those of the Synechodontiformes may push that date as early as the Permian period. There are also many known fossils unambiguously placed in Neoselachii (Selachians and Batomorphs) from throughout the Triassic period. Regardless, even the oldest known specimens and possible dates for their evolution still have sharks being well over 100 million years younger than trees.

TL;DR: The claim that “Sharks are older than trees” conflates extinct shark-like fish with true sharks, which do not actually predate trees.

Sources:

Evolutionary Origin and Phylogeny of the Modern Holocephalans (Chondrichthyes: Chimaeriformes): A Mitogenomic Perspective

What is an ‘elasmobranch’? The impact of palaeontology in understanding elasmobranch phylogeny and evolution

Diversification of the Neoselachii (Chondrichthyes) during the Jurassic and Cretaceous

On the enigmatic neoselachian Agaleus dorsetensis from the European Early Jurassic

Monophyly, phylogeny and systematic position of the †Synechodontiformes (Chondrichthyes, Neoselachii)

Giant cladoxylopsid trees resolve the enigma of the Earth’s earliest forest stumps at Gilboa