I'm not sure but I suspect your box is using HDCP and unfortunately it would appear that the Slingbox 500 cannot deal with a HDCP-protected HDMI input, so you'd still need to connect the analogue cables from the TiVo to the Slingbox - but as you say, this isn't working when HDMI is plugged in.
Try searching the support section on this site for "HDCP" and there seem to be a few articles relating to the 500:
Another option would be to ditch HDMI altogether. If your TiVo has component outputs then just use these, and then use the component pass through into the TV. Personally I cannot tell the difference between HDMI and Component for a 1080i/720p broadcast. If you have sufficient upload bandwidth you'll also get a far better picture when viewing your box with a Slingplayer using this method.
Your TiVo may not have component outputs though, in which case you'd have to buy an HDMI-to-Component converter like the HDFury or another one that can do the conversion from a HDCP-protected source.
The problem is related to the HDCP encryption of the HDMI signal. When your TV is switched on it is detected by TiVo DVR as a device, which supports HDCP. When the TV is switched off its HDCP module is also switched off and TiVo DVR recognizes then your TV as a non-HDCP compliant device. In the result TiVo DVR blocks the video signal on all its outputs.. It is why your slingbox does not see any signal from TiVo DVR, when the TV is switched off.
The problem would not exist if you use an iHDMI cable (iHDMI.eu) instead of the passive HDMI cable.