Given that a dining room is a place for gathering and sharing, "cozy" seems to me like the ultimate adjective and goal for this type of space.
In some cases, paint or wall-coverings do the heavy lifting — consider the moody blue in Image 2, the encompassing black in Image 6, and warm red in Image 10. Otherwise, including details such as wood tones, textiles, upholstery, and rugs can bring warmth and coziness to a space.
1. Pete & James's Natural & Neon Loft
2. Michele & Ryan Tansey's Antiquarian Hideaway
3. Peter & Blaire's "Bohemian Preppy" Seattle Abode
4. Celeste & Nicolai's Modern Art Loft
5. Nancy, Ethan & Ohle's 1903 Cabin in the Country
6. Hallie & Mark's Hip Slice of History
7. Meredith's Historic Farmhouse Dream
8. Johnelle & Brian's Rejuvenated Treasures Home
9. Dana's Bungalow Beautification
10. Mike's Fantasy Addition in the Sky
Images: As linked above.