Like Shirlock said, this does happen. I have taped a few staple mishaps along the way. Each incident is evaluated differently. If it is a neutral with just the insulation rubbed away (not a compromise to the conductor!). And I was also sure that it was used as a neutral (and not a switched lead). Then I have layed the insulation back and given it a few wraps with the Scotch 33 (600V insulation per wrap!, not the cheap stuff) and felt confident it was safe. Is that code compliant? Will an inspector fail the job if he saw it? Absolutely! The inspector does not know what is behind the tape and is not going to take your word for it.
Okay, so obviously you want to sleep at night so here are a few code compliant ideas.
- Add a "deep Handy box" (single outlet box) at outlet height and splice the wire coming from the floor to a new wire going up if you can replace that wire. If the wire is a feed, perhaps include an outlet in the box. If not, just put a blank cover on it. As long as you can get to the box, a proper splice will be code compliant.
- Can't add a box there? How about the other side of the wall? Maybe you will even get lucky and end up in a closet! Only if you're in a closet, you must make it a J-box w/blank cover, outlets, switches are not allowed in closets.
EDIT:
I was told by a licensed Electrician awhile ago that it is a code violation to put outlets or switches inside the closet. I never questioned that because all new construction I have seen here in many years has the light switch outside of the door! It made sense, with posible arcing from a switch and clothes and all. I'm thinking it may be a local code requirement. I know he also talked about how you need to do it as the inspector wanted, in addition to NEC code. I did find however that code clearly states that no electrical breaker panels can be in a clothes closet, perhaps its a variation of that?. I have to stand by my friends interpretation, I know he is competent Electrician.
Without being able to see the cables as they enter the cabinet; or the ability to touch or trace them, here is what I assume is going on.
Definitions:

Grounded (neutral) from the service
A typical single split phase service is made up of 3 wires. Two ungrounded (hot) conductors, and one grounded (neutral) conductor. The ungrounded (hot) conductors will connect to the main service panel through a disconnect (usually a large breaker), while the grounded (neutral) connects to the neutral lug. The neutral lug will be bonded (electrically connected) to the neutral bus bar, and all grounded (neutral) branch circuit conductors will terminate at the neutral bus.
Grounding Electrode Conductor
This conductor is used to connect the grounding electrode (ground rod, etc.), to the grounding bus in the panel. All equipment grounding conductors will be connected to this bus.
Bonding Jumper
The bonding jumper is used to bond (electrically connect), the un-energized metal parts of the panel to the grounding system.
Assumption:
Since it appears that (what I assume is) the grounding electrode conductor terminates at the neutral bus, I'm also assuming that this is the main service disconnect. This leads me to believe that the neutral and grounding buses are bonded (electrically connected). In which case, technically, grounded (neutral) branch circuit conductors can terminate at the grounding bus.
So you have two options:
Terminate the grounded (neutral) from the new circuit to the grounding bus.
Move the green wire that is terminated on the neutral bus, to the grounding bus. Then terminate the grounded (neutral) from the new circuit, to the freed up slot on the neutral bus.
Additional Information and Code Compliance:
Number of Conductors
Since this is a new circuit, it has to be installed to current code standards.
National Electrical Code 2011
ARTICLE 250 — GROUNDING AND BONDING
VI. Equipment Grounding and Equipment Grounding Conductors
250.140 Frames of Ranges and Clothes Dryers. Frames of electric ranges, wall-mounted ovens, counter-mounted cooking units, clothes dryers, and outlet or junction boxes that are part of the circuit for these appliances shall be connected to the equipment grounding conductor in the manner specified by 250.134 or 250.138.
Which in this case means installing a NEMA 14 receptacle for the dryer, and a proper grounding conductor.

You'll have to follow the dryer manufacturers installation instructions for upgrading to a 4 wire cord. For more information see this answer, and this answer.
Since you've said that you're already using 4 wire cable, you'll simply have to terminate the grounding conductor in the cable to the grounding bus in the service panel. Then connect the other end of the grounding conductor to the grounding terminal in the dryer receptacle.
Size of Conductors
You'll also want to be sure that you're using the proper size breaker and conductors. In the case of a dryer, you'll typically use a 30 ampere breaker and 10 AWG conductors (depending on the length of the run). However, you'll want to check the dryer manufacturers installation instructions to verify this.
Best Answer
The "tar" is not a concern, it sounds like the box you described is an old work box they are listed and legal, they are held in by small stops or wings that hold the box to the sheetrock.