DMA IRQHandler not working

It's not getting to the IRQ because you're not triggering any DMA transfers. The TIM settings have no relationship to the desired triggering of the DMA.

The RM0008 Reference Manual might prove more useful than Google.

OK so I came to the conclusion that I was using the wrong DMA channel, and that there was a timer practically dedicated for the DAC. Still the main problem persists, I can't enter in the IRQHandler. The DMA and DAC are working fine now.

/* Includes ------------------------------------------------------------------*/
#include "stm32f10x.h"
#include "LED.h"

/* Private typedef -----------------------------------------------------------*/
/* Private define ------------------------------------------------------------*/

/* Init Structure definition */
DAC_InitTypeDef            DAC_InitStructure;
DMA_InitTypeDef            DMA_InitStructure;
NVIC_InitTypeDef                                         NVIC_InitStructure;

/* Private macro -------------------------------------------------------------*/
/* Private variables ---------------------------------------------------------*/
uint8_t Escalator8bit[256];

/* Private functions ---------------------------------------------------------*/

void RCC_Configuration(void)
{
  /* Enable peripheral clocks ------------------------------------------------*/
#if !defined STM32F10X_LD_VL && !defined STM32F10X_MD_VL
  /* DMA2 clock enable */
  RCC_AHBPeriphClockCmd(RCC_AHBPeriph_DMA2, ENABLE);
#else
  /* DMA1 clock enable */
  RCC_AHBPeriphClockCmd(RCC_AHBPeriph_DMA1, ENABLE);
#endif
  /* GPIOA Periph clock enable */
  RCC_APB2PeriphClockCmd(RCC_APB2Periph_GPIOA, ENABLE);
  /* DAC Periph clock enable */
  RCC_APB1PeriphClockCmd(RCC_APB1Periph_DAC, ENABLE);
  /* TIM6 Periph clock enable */
  RCC_APB1PeriphClockCmd(RCC_APB1Periph_TIM6, ENABLE);
}

void GPIO_Configuration(void)
{
  GPIO_InitTypeDef GPIO_InitStructure;

  /* Once the DAC channel is enabled, the corresponding GPIO pin is automatically
     connected to the DAC converter. In order to avoid parasitic consumption,
     the GPIO pin should be configured in analog */
  GPIO_InitStructure.GPIO_Pin =  GPIO_Pin_4;
  GPIO_InitStructure.GPIO_Mode = GPIO_Mode_AIN;
  GPIO_Init(GPIOA, &GPIO_InitStructure);
}

void TIMER_Configuration(){
  /* TIM6 Configuration */
  TIM_PrescalerConfig(TIM6, 0xF, TIM_PSCReloadMode_Update);
  TIM_SetAutoreload(TIM6, 0xFF);
  /* TIM6 TRGO selection */
  TIM_SelectOutputTrigger(TIM6, TIM_TRGOSource_Update);
}

void DAC_Configuration(){
  /* DAC channel1 Configuration */
  DAC_InitStructure.DAC_Trigger = DAC_Trigger_T6_TRGO;
  DAC_InitStructure.DAC_WaveGeneration = DAC_WaveGeneration_None;
  DAC_InitStructure.DAC_OutputBuffer = DAC_OutputBuffer_Disable;
  DAC_Init(DAC_Channel_1, &DAC_InitStructure);
}

void DMA_Configuration(){
#if !defined STM32F10X_LD_VL && !defined STM32F10X_MD_VL
  /* DMA2 channel3 configuration */
  DMA_DeInit(DMA2_Channel3);
#else
  /* DMA1 channel3 configuration */
  DMA_DeInit(DMA1_Channel3);
#endif

  DMA_InitStructure.DMA_PeripheralBaseAddr = (uint32_t)&DAC->DHR8R1;
  DMA_InitStructure.DMA_MemoryBaseAddr = (uint32_t)&Escalator8bit;
  DMA_InitStructure.DMA_DIR = DMA_DIR_PeripheralDST;
  DMA_InitStructure.DMA_BufferSize = 256;
  DMA_InitStructure.DMA_PeripheralInc = DMA_PeripheralInc_Disable;
  DMA_InitStructure.DMA_MemoryInc = DMA_MemoryInc_Enable;
  DMA_InitStructure.DMA_PeripheralDataSize = DMA_PeripheralDataSize_Byte;
  DMA_InitStructure.DMA_MemoryDataSize = DMA_MemoryDataSize_Byte;
  DMA_InitStructure.DMA_Mode = DMA_Mode_Circular;
  DMA_InitStructure.DMA_Priority = DMA_Priority_High;
  DMA_InitStructure.DMA_M2M = DMA_M2M_Disable;

#if !defined STM32F10X_LD_VL && !defined STM32F10X_MD_VL
  DMA_Init(DMA2_Channel3, &DMA_InitStructure);
  /* Enable DMA2 Channel3 */
  DMA_Cmd(DMA2_Channel3, ENABLE);
#else
  DMA_Init(DMA1_Channel3, &DMA_InitStructure);
  /* Enable DMA1 Channel3 */
  DMA_Cmd(DMA1_Channel3, ENABLE);
#endif

  DMA_ITConfig(DMA1_Channel3 , DMA_IT_TC | DMA_IT_HT , ENABLE);
}

void NVIC_Configuration(){
   //Enable DMA1 channel IRQ Channel
   NVIC_InitStructure.NVIC_IRQChannel = DMA1_Channel3_IRQn;
   NVIC_InitStructure.NVIC_IRQChannelPreemptionPriority = 0;
   NVIC_InitStructure.NVIC_IRQChannelSubPriority = 0;
   NVIC_InitStructure.NVIC_IRQChannelCmd = ENABLE;
   NVIC_Init(&NVIC_InitStructure);
}

void DMA1_Channel3_IRQHandler(void){
   LED_On(0);
   if(DMA_GetITStatus(DMA1_IT_TC3))
  {
      LED_On(1);
   //Clear DMA1 Channel1 Half Transfer, Transfer Complete and Global interrupt pending bits
    DMA_ClearITPendingBit(DMA1_IT_GL3);
  }
}

int main(void)
{
                int i;
                for(i=0; i<256; i++){
                        Escalator8bit[i] = i;
                }
  /*!< At this stage the microcontroller clock setting is already configured,
       this is done through SystemInit() function which is called from startup
       file (startup_stm32f10x_xx.s) before to branch to application main.
       To reconfigure the default setting of SystemInit() function, refer to
       system_stm32f10x.c file
     */

  /* System Clocks Configuration */
  RCC_Configuration();
        LED_Initialize();
  /* Once the DAC channel is enabled, the corresponding GPIO pin is automatically
     connected to the DAC converter. In order to avoid parasitic consumption,
     the GPIO pin should be configured in analog */
  GPIO_Configuration();
        TIMER_Configuration();
        DAC_Configuration();
        DMA_Configuration();
        NVIC_Configuration();
  /* Enable DAC Channel1: Once the DAC channel1 is enabled, PA.04 is
     automatically connected to the DAC converter. */
  DAC_Cmd(DAC_Channel_1, ENABLE);
  /* Enable DMA for DAC Channel1 */
  DAC_DMACmd(DAC_Channel_1, ENABLE);
  /* TIM6 enable counter */
  TIM_Cmd(TIM6, ENABLE);
        LED_On(2);
  while (1)
  {
  }
}

Pretty sure it needs to be DMA2_Channel3. See RM0008 Rev15

Don't enable the HT interrupt unless you plan to service it.

And if you're using C++ (.cpp) then

extern "C" void DMA2_Channel3_IRQHandler(void)
{
..
}

OK indeed! thanks a lot for your help!