I go and inspect and there is literally no leaks upstairs, no water lines in this section of the house and no rain for days outside and no sign of moisture intrusion from outside.

Is the spot next to a sliding door where moisture could be an issue?

Inspection revealed improperly shimmied joists and structural sagging, furnaces were 31+ years old (seller would not replace), moisture behind some of the shower tiling, chimney need mason work done and looked structurally suspect.

The money he thinks he is saving by doing it himself may be eaten up in extra repairs inside due to moisture damage. 

labor/Material rough estimateFlooring(labor) : 2500Flooring Material: 1.5 per sq feet + paddingKitchen - 3000Cabinet & counter top ( see/select home depot website)Bathroom - 3000Tile + Tub and fixtures ( see/select home depot website )Electric - 1500 ( inside unit )Electric Materials ( see/select home depot websitePlumbing - 1500 ( inside unit kitchen and bathrooms )Plumbing Supplies see/select home depot websitePainting - 3000Paint see/select home depot websiteWindows ( replacements ) per window $450 all includedCleaning - $350 final cleaning Trash Out -$500 ( includes dumpster and dumping fee )Appliances - 2000// Total (without materials): 18’700 From Inspection Report Labor ONLY 3.3 - Repair gutters - $475 5.4 - Exhaustive fan (kitchen) - $3505.7 - Dishwasher - replace ( $450 )9.3 - water heater - fixed and fixed all the leaky gas lines - 2K10.7 - replace all outlets - $25002.9 - Insulation not present (basement) - 4.1 Damaged drywall (wallboard contractor) - $ 7504.2 - Wall - elevated moisture leaking window - replace window ?

The reason I ask is because I'm not sure if there's something we'd need to do to prevent moisture from building up.

Looks like you have a moisture issue, and that will put off buyers.

In addition, we rerouted the airways to reduce moisture and make it more efficient.

Install something for moisture meter in basement. 

Our lease statesIn signing this Lease, Tenant has examined the Premises and certifies that Tenant has not observed mold, mildew or moisture within the Premises, unless otherwise noted during the move-in walkthrough.