Has there ever been a case where we have found what appears to be a prehistoric fossil but have no way of dating it?
The short answer is no, because we always have some constraint of how old something is.  Sometimes we have fossils with large date error bars, and often we do not know for how long that species existed. A large date range constraint in some cases, like archeology, might not be useful. If the specimen is taken without noting the location it was taken from, then we have no context to date it, which is a big reason not to plunder paleontological sites.

For dating a fossil, the context of the rock layer, other fossils, cross-cutting units, and location is key.  Geologists must use a combination of absolute and relative age dating techniques.  Geologists examine the rock layer from which the fossil was found and compare that to other rock layers that stratigraphically correlate to it.  They use basic relative dating approaches, including:


Principle of Superposition: older layers on bottom, younger layers on top
Principle of Cross Cutting Relationships: if a feature (like igneous intrusions and faults) cuts through a layer, then that feature is younger than the layer
Principle of Original Horizontality: Layers are deposited in a horizontal fashion
Principle of Lateral Continuity: Layers are general deposited in a laterally continuous manner until they pinch out; if you see a canyon and similar layer on either side, you can assume it was previously continuous and the canyon is younger.
Principle of Faunal Succession: The presence or absence groups of fossils can be used to hone in on the time period


There are a wide variety of absolute dating techniques, each with its own time and medium constraints. Radiometric dating, fission track dating, and paleomagnetic approaches are typically restricted to igneous rocks. For radiometric dating, only a small quantity of isotopes are needed for a measurable date. Igneous rocks contain usable quantities of minerals like zircon that are good at holding atoms of uranium and other elements useful in for the absolute age-dating technique. Geologists use absolute dates of rocks that can be dated in combination with relative dating techniques to constrain the age of a rock layer that cannot be dated with absolute techniques.  

Fossil morphology must also be considered to determine if you are really looking at a new species or just a special case of a previously identified species. Sometimes, evolutionary approaches can be applied to help place the specimen into a timeline.

Figuring out the temporal extent of a species takes more time and multiple discoveries of samples of that species to better constrain that extent.

A combination of approaches and techniques can help paleontologists be detectives in determining the age of specimen. Like other sciences, geology is social in the sense that it relies on the culmination of the work of many, with multiple lines of complementary pieces of information, to come to a conclusive age. 

Here are some great articles related to your question:


DiMaggio, E. and Tripathy-Lang, A., 2019, How Do Scientists Date Fossils?: Smithsonian Magazine, online: https://www.smithsonianmag.com/smithsonian-institution/how-do-scientists-date-fossils-180972391/
Peppe, D. J. and Deino, A. L., 2013, Dating Rocks and Fossils Using Geologic Methods: Nature Education Knowledge vol. 4, no. 10:1, online: https://www.nature.com/scitable/knowledge/library/dating-rocks-and-fossils-using-geologic-methods-107924044/


Here is an example of an "undatable" fossil, where the age could "only" be constrained by 2 million years:


Yong, E., 2015, Why Don't We Know the Age of the New Ancient Human?: The Atlantic, online: https://www.theatlantic.com/science/archive/2015/09/why-dont-we-know-the-age-of-the-new-human-ancestor-homo-naledi/405148/