do I need a new DVR box for the new 4k television I purchased

That doesn't make sense. HDMI is HDMI. All HDMI cables are the same. You don't mention what cable box you have. You can find the model number on the X1 about screen, or on the bottom of the cable box. The only 4K capable boxes are the XG1v4 and Xi6. Other boxes will work, but will max out at standard HD. Also, the only 4K available on Comcast is some OnDemand movies and sports. There are no linear channels in 4K. All linear TV is 720p.

There may be differences in capabilities with higher end resolutions, but using an older cable may just mean 4K may not work or be limited. There may also be special connectors like mini-HDMI, but TV's and most any TV devices would use standard HDMI.