I found this logo I really like. How can I achieve this effect in Photoshop or Illustrator?
Referring to the given solution in Illustrator: The same is possible in Photoshop. You can right click a text layer in the layers panel and select Warp Text. There's available the flag warping.
Then tou can add the radial blur. This will ask you to allow rasterizing the text to make the blurring possible.
Keep a copy of the text in a hidden layer to be able to make edits later.