With modern OSs, practically none. Linus Torvalds reportedly said its task is to "just load the OS and get the hell out of there".
Older operating systems like MS-DOS relied on the BIOS for many tasks (e.g. disk access), by calling interrupts.
With modern OSs, the bootloader quickly switches into 32- or 64-bit mode and executes the OS kernel. The kernel can register its own interrupt handlers, which can be called by user-space applications. The kernel's routines can be more portable (since they don't depend on the specific hardware), more flexible (OS vendors can change them on demand rather than having to use whatever came with the hardware), more sophisticated (they can execute arbitrarily complex code rather than what was programmed into the BIOS), and more secure (since the OS can control access to shared resources and prevent programs from clobbering each other, implementing its own arbitrary permissions schemes).
To interact with specific hardware, OSs can load and use its own device drivers. So there's no need for the OS or applications to call most BIOS routines at all. In fact, for security reasons, BIOS interrupts are even disabled. Since the BIOS lives in 16-bit real mode it's harder to call for modern OSs.
While use of the BIOS is very limited while the OS runs, its functions are still peripherally used. For example, when a computer sleeps, the OS is not running and it ultimately falls to the firmware to set the hardware to the correct state to pause and resume the OS. These uses are generally limited to ACPI calls rather than calls to the full BIOS interface. ACPI is a BIOS extension that "brings power management under the control of the operating system (OSPM), as opposed to the previous BIOS-central system, which relied on platform-specific firmware to determine power management and configuration policy".
Note that officially "BIOS" refers to a particular firmware interface, but the term is commonly used to refer to computer firmware in general. Some recent computers (especially Apple ones) have replaced BIOS (sensu strictu) with UEFI, which of course then is what is called to implement these functions.
For more information about how the role of the BIOS has diminished over time, see Wikipedia.