Nonnegative impulse response (NNIR) filters have found many applications in signal processing and information fusion areas. Evidence filtering is one of the examples among others. An evidence filter is required to satisfy a nonnegativity condition and a normalization condition on its impulse response coefficients, and thus is basically an NNIR filter. This paper considers the design of nonnegative finite impulse response (FIR) filters based on frequency response approximation and proposes a constrained minimax design formulation using the fundamental limitations on the NNIR filter's frequency responses recently developed in the literature. The formulation is converted into a linearly constrained positive-definite quadratic programming and then solved with the Goldfarb-Idnani algorithm. The proposed method is applicable to nonnegative FIR lowpass as well as other types of filters. Design examples demonstrate the effectiveness of the proposed method.