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I downloaded a jessie raspbian image and it shows two partitions.

[Michael@devserver ~]$ fdisk -l 2017-07-05-raspbian-jessie-lite.img
You must set cylinders.
You can do this from the extra functions menu.

Disk 2017-07-05-raspbian-jessie-lite.img: 0 MB, 0 bytes
255 heads, 63 sectors/track, 0 cylinders
Units = cylinders of 16065 * 512 = 8225280 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disk identifier: 0xa8790229

 Device Boot Start End Blocks Id System
2017-07-05-raspbian-jessie-lite.img1 1 6 42702+ c W95 FAT32 (LBA)
Partition 1 does not end on cylinder boundary.
2017-07-05-raspbian-jessie-lite.img2 6 210 1638081 83 Linux
Partition 2 does not end on cylinder boundary.
[Michael@devserver ~]$

I copied the image to a SD which had previously been used (sorry, I don't have the fdisk output before copying the image):

[Michael@devserver ~]$ sudo dd bs=4M if=2017-07-05-raspbian-jessie-lite.img of=/dev/sdf
411+1 records in
411+1 records out
1725629563 bytes (1.7 GB) copied, 149.961 s, 11.5 MB/s
[Michael@devserver ~]$

Inspecting the device now shows the following:

[Michael@devserver ~]$ sudo fdisk -l /dev/sdf

Disk /dev/sdf: 31.9 GB, 31914983424 bytes
64 heads, 32 sectors/track, 30436 cylinders
Units = cylinders of 2048 * 512 = 1048576 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disk identifier: 0xa8790229

 Device Boot Start End Blocks Id System
/dev/sdf1 5 46 42702+ c W95 FAT32 (LBA)
Partition 1 does not end on cylinder boundary.
/dev/sdf2 47 1646 1638081 83 Linux
Partition 2 does not end on cylinder boundary.

[Michael@devserver ~]$ lsblk -f /dev/sdf
NAME FSTYPE LABEL UUID MOUNTPOINT
sdf
├─sdf1
├─sdf2
├─sdf5
├─sdf6
└─sdf7
[Michael@devserver ~]$ cat /proc/partitions | grep sdf
 8 80 31166976 sdf
 8 81 1243951 sdf1
 8 82 1 sdf2
 8 85 32767 sdf5
 8 86 67584 sdf6
 8 87 29815296 sdf7
[Michael@devserver ~]$ sudo blkid /dev/sdf*
/dev/sdf1: LABEL="boot" UUID="109A-9113" TYPE="vfat"
[Michael@devserver ~]$

Why do partitions sdf5, sdf6, and sdf7 still exist? Should they be removed, and if so, how?

After copying the image, did you run partprobe /dev/sdf or eject & re-insert the card? If you didn't, the kernel might be still using the old partition table for the card.

When you overwrite an entire disk/SD card, you'll need to tell the kernel that it should re-read the partition table. partprobe is the easiest command for doing it.