Many things don't work well together: oil and water, drinking and driving, and as it turns out, Samsung's TecTiles and the Galaxy S4. Samsung is acknowledging that the current line of NFC tags cannot be read by its new flagship phone. In fact, even the official TecTile site's Buy Now link is now defunct.
The SGS4 incorporates a Broadcom BCM2079x chip which lacks support for some types of MIFARE tags from NXP Semiconductors, including TecTiles which are built with MIFARE Classic chips. Samsung is planning to launch TecTiles 2 to replace the older stickers. Here is their official statement given to Android Police: