In this paper, we introduce a video-based wildfire detection scheme based on a computationally efficient additive deep neural network, which we call AddNet. This AddNet is based on a multiplication-free vector operator, which performs only addition and sign manipulation operations. In this regard, we construct a dot product-like operation from the mf-operator and use it to define dense and convolutional feed-forwarding passes in AddNet. We train AddNet on images taken from forestry surveillance cameras. Our experiments show that AddNet can achieve a time-saving by 12.4% when compared to an equivalent regular convolutional neural network (CNN). Furthermore, the smoke recognition performance of AddNet is as good as regular CNNs and substantially better than binary-weight neural networks.