Motivated by the definition of Tribonacci quaternions, we define hyper-dual numbers whose components involve Tribonacci and Tribonacci-Lucas numbers. We refer to these new numbers as hyper-dual Tribonacci numbers and hyper-dual Tribonacci-Lucas numbers, respectively. In this paper, we also establish some properties of these numbers and present useful identities involving them. Furthermore, we investigate formulas for the generalized sum and the sum with alternating signs for Tribonacci and Tribonacci-Lucas numbers using a new method. Based on these results, we derive the corresponding formulas for the generalized and alternating sign sums of hyper-dual Tribonacci and hyper-dual Tribonacci-Lucas numbers.